################################################################### # # Code for finding Latin squares that admit automorphisms # and autotopisms. On my computer, install by typing: # # Read("/home/dougy/Dropbox/Thesis/Thoughts/autt5.txt"); # # Replace the filename above with the filename of this file # on your computer. # # - Douglas S. Stones # ################################################################### # # A selection of key features: # # # IsAutotopismLatinSquare(L,alpha,beta,gamma) # returns true of L is a Latin square and (alpha,beta,gamma) # is an autotopism of L. # # IsAutomorphismLatinSquare(L,alpha) # returns true of L is a Latin square and alpha is an automorphism of L. # # LatinSquareForAutomorphismWithCycleDAndFFixedPoints(d,f) # returns a Latin square with automorphism alpha # where alpha has the cycle structure [d,1,1,...,1] (with f 1's) # returns fail if impossible # # LatinSquareForAutomorphismWithTwoCycles(d1,d2,f) # returns a Latin square with automorphism alpha # where alpha has the cycle structure [d1,d2,1,1,...,1] (with f 1's) # returns fail if impossible # (assumes d1>=d2) # # LatinSquareForAutomorphismWithThreeCycles(d1,d2,d3,f) # returns a Latin square with automorphism alpha # where alpha has the cycle structure [d1,d2,d3,1,1,...,1] (with f 1's) # returns fail if impossible # (assumes d1>=d2>=d3) # # LatinSquareForAutomorphismWithM_DCycles(m,d,f) # returns a Latin square with automorphism alpha # where alpha has the cycle structure [d,d,..,d] (with m d-cycles) # returns fail if impossible # # AutRep[n] contains a list of autotopisms of Latin squares of order n # the list is exhaustive up to cycle structure # LatinRep[n][j] is a Latin square that admits the autotopism AutRep[n][j] # # # Note: All row/column/symbol indices are {1,2,..,n} # # # Sample usage: # # gap> L:=LatinSquareForAutomorphismWithThreeCycles(12,4,3,1); # [ [ 20, 8, 3, 13, 4, 14, 5, 15, 6, 19, 7, 17, 9, 18, 12, 10, 16, 11, 1, 2 ], # [ 18, 20, 9, 4, 14, 5, 15, 6, 16, 7, 17, 8, 11, 10, 19, 1, 2, 13, 12, 3 ], # [ 9, 19, 20, 10, 5, 15, 6, 16, 7, 13, 8, 18, 2, 12, 11, 17, 1, 3, 14, 4 ], # [ 19, 10, 17, 20, 11, 6, 16, 7, 13, 8, 14, 9, 18, 3, 1, 12, 15, 2, 4, 5 ], # [ 10, 17, 11, 18, 20, 12, 7, 13, 8, 14, 9, 15, 1, 19, 4, 2, 5, 16, 3, 6 ], # [ 16, 11, 18, 12, 19, 20, 1, 8, 14, 9, 15, 10, 3, 2, 17, 5, 4, 6, 13, 7 ], # [ 11, 13, 12, 19, 1, 17, 20, 2, 9, 15, 10, 16, 6, 4, 3, 18, 14, 5, 7, 8 ], # [ 13, 12, 14, 1, 17, 2, 18, 20, 3, 10, 16, 11, 19, 7, 5, 4, 8, 15, 6, 9 ], # [ 12, 14, 1, 15, 2, 18, 3, 19, 20, 4, 11, 13, 5, 17, 8, 6, 7, 9, 16, 10 ], # [ 14, 1, 15, 2, 16, 3, 19, 4, 17, 20, 5, 12, 7, 6, 18, 9, 13, 8, 10, 11 ], # [ 1, 15, 2, 16, 3, 13, 4, 17, 5, 18, 20, 6, 10, 8, 7, 19, 11, 14, 9, 12 ], # [ 7, 2, 16, 3, 13, 4, 14, 5, 18, 6, 19, 20, 17, 11, 9, 8, 10, 12, 15, 1 ], # [ 6, 18, 5, 8, 10, 19, 9, 12, 2, 17, 1, 4, 20, 16, 14, 13, 3, 7, 11, 15 ], # [ 5, 7, 19, 6, 9, 11, 17, 10, 1, 3, 18, 2, 14, 20, 13, 15, 12, 4, 8, 16 ], # [ 3, 6, 8, 17, 7, 10, 12, 18, 11, 2, 4, 19, 16, 15, 20, 14, 9, 1, 5, 13 ], # [ 17, 4, 7, 9, 18, 8, 11, 1, 19, 12, 3, 5, 15, 13, 16, 20, 6, 10, 2, 14 ], # [ 15, 5, 4, 14, 8, 7, 13, 11, 10, 16, 2, 1, 12, 9, 6, 3, 19, 20, 17, 18 ], # [ 2, 16, 6, 5, 15, 9, 8, 14, 12, 11, 13, 3, 4, 1, 10, 7, 18, 17, 20, 19 ], # [ 4, 3, 13, 7, 6, 16, 10, 9, 15, 1, 12, 14, 8, 5, 2, 11, 20, 19, 18, 17 ], # [ 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 18, 19, 20 ] ] # gap> alpha:=(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)(17,18,19); # (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)(17,18,19) # gap> IsAutomorphismLatinSquare(L,alpha); # true # gap> # # # gap> L:=LatinRep[16][100]; # [ [ 16, 13, 2, 12, 14, 9, 8, 10, 15, 4, 6, 1, 3, 11, 7, 5 ], # [ 9, 16, 14, 3, 11, 13, 10, 5, 2, 15, 1, 7, 12, 4, 8, 6 ], # [ 4, 10, 16, 13, 6, 12, 14, 11, 8, 3, 15, 2, 1, 9, 5, 7 ], # [ 14, 1, 11, 16, 12, 7, 9, 13, 3, 5, 4, 15, 10, 2, 6, 8 ], # [ 5, 15, 8, 10, 16, 14, 6, 1, 13, 2, 12, 3, 4, 7, 11, 9 ], # [ 11, 6, 15, 5, 2, 16, 13, 7, 4, 14, 3, 9, 8, 1, 12, 10 ], # [ 6, 12, 7, 15, 8, 3, 16, 14, 10, 1, 13, 4, 2, 5, 9, 11 ], # [ 15, 7, 9, 8, 13, 5, 4, 16, 1, 11, 2, 14, 6, 3, 10, 12 ], # [ 7, 14, 6, 1, 9, 15, 12, 3, 16, 13, 10, 5, 11, 8, 4, 2 ], # [ 2, 8, 13, 7, 4, 10, 15, 9, 6, 16, 14, 11, 5, 12, 1, 3 ], # [ 8, 3, 5, 14, 10, 1, 11, 15, 12, 7, 16, 13, 9, 6, 2, 4 ], # [ 13, 5, 4, 6, 15, 11, 2, 12, 14, 9, 8, 16, 7, 10, 3, 1 ], # [ 1, 4, 3, 2, 7, 6, 5, 8, 9, 12, 11, 10, 16, 15, 14, 13 ], # [ 3, 2, 1, 4, 5, 8, 7, 6, 11, 10, 9, 12, 15, 16, 13, 14 ], # [ 10, 11, 12, 9, 3, 4, 1, 2, 7, 8, 5, 6, 14, 13, 15, 16 ], # [ 12, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 16, 15 ] ] # gap> AutRep[16][100]; # [ (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14), # (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14) ] # gap> alpha:=AutRep[16][100][1];; # gap> beta:=AutRep[16][100][2];; # gap> gamma:=AutRep[16][100][3];; # gap> IsAutotopismLatinSquare(L,alpha,beta,gamma); # true # gap> # # # gap> L:=LatinSquareForM_DCycles(2,4,0); # [ [ 5, 1, 6, 4, 7, 3, 8, 2 ], # [ 1, 6, 2, 7, 3, 8, 4, 5 ], # [ 8, 2, 7, 3, 6, 4, 5, 1 ], # [ 4, 5, 3, 8, 2, 7, 1, 6 ], # [ 7, 3, 8, 2, 5, 1, 6, 4 ], # [ 3, 8, 4, 5, 1, 6, 2, 7 ], # [ 6, 4, 5, 1, 8, 2, 7, 3 ], # [ 2, 7, 1, 6, 4, 5, 3, 8 ] ] # gap> alpha:=CycleStructureToPermutation([4,4]); # (1,2,3,4)(5,6,7,8) # gap> IsAutomorphismLatinSquare(L,alpha); # true # gap> # ################################################################### # # Small Latin squares data # # AutRep[n][j] is a representative from the j-th equivalence class in Aut(n) # LatinRep[n][j] is a Latin square that admits the autotopism AutRep[n][j] # AutRep:=[]; LatinRep:=[]; AutRep[1]:=[[(),(),()]]; LatinRep[1]:=[[[1]]]; AutRep[2]:=[[(),(),()],[(),(1,2),(1,2)]]; LatinRep[2]:=[[[1,2],[2,1]],[[1,2],[2,1]]]; AutRep[3]:=[[(),(),()],[(),(1,2,3),(1,2,3)],[(1,2),(1,2),(1,2)],[(1,2,3),(1,2,3),(1,2,3)]]; LatinRep[3]:=[[[3,2,1],[1,3,2],[2,1,3]],[[1,2,3],[2,3,1],[3,1,2]],[[1,3,2],[3,2,1],[2,1,3]],[[1,3,2],[3,2,1],[2,1,3]]]; AutRep[4]:=[[(),(),()],[(),(1,2)(3,4),(1,2)(3,4)],[(),(1,2,3,4),(1,2,3,4)],[(1,2),(1,2),(1,2)],[(1,2),(1,2)(3,4),(1,2)(3,4)],[(1,2),(1,2,3,4),(1,2,3,4)],[(1,2)(3,4),(1,2)(3,4),(1,2)(3,4)],[(1,2)(3,4),(1,2,3,4),(1,2,3,4)],[(1,2,3),(1,2,3),(1,2,3)]]; LatinRep[4]:=[[[2,1,3,4],[3,4,2,1],[4,2,1,3],[1,3,4,2]],[[1,2,3,4],[2,1,4,3],[3,4,1,2],[4,3,2,1]],[[1,2,3,4],[2,3,4,1],[3,4,1,2],[4,1,2,3]],[[3,4,1,2],[4,3,2,1],[1,2,3,4],[2,1,4,3]],[[1,3,2,4],[4,2,3,1],[2,1,4,3],[3,4,1,2]],[[1,4,3,2],[3,2,1,4],[2,3,4,1],[4,1,2,3]],[[1,3,2,4],[4,2,3,1],[2,4,1,3],[3,1,4,2]],[[1,4,3,2],[3,2,1,4],[2,1,4,3],[4,3,2,1]],[[1,3,4,2],[4,2,1,3],[2,4,3,1],[3,1,2,4]]]; AutRep[5]:=[[(),(),()],[(),(1,2,3,4,5),(1,2,3,4,5)],[(1,2)(3,4),(1,2)(3,4),(1,2)(3,4)],[(1,2,3),(1,2,3),(1,2,3)],[(1,2,3,4),(1,2,3,4),(1,2,3,4)],[(1,2,3,4,5),(1,2,3,4,5),(1,2,3,4,5)]]; LatinRep[5]:=[[[4,5,3,1,2],[1,3,5,2,4],[5,1,2,4,3],[2,4,1,3,5],[3,2,4,5,1]],[[1,2,3,4,5],[2,3,4,5,1],[3,4,5,1,2],[4,5,1,2,3],[5,1,2,3,4]],[[1,3,2,5,4],[4,2,5,1,3],[3,5,1,4,2],[5,4,3,2,1],[2,1,4,3,5]],[[1,4,5,2,3],[5,2,4,3,1],[4,5,3,1,2],[2,3,1,4,5],[3,1,2,5,4]],[[1,3,2,5,4],[5,2,4,3,1],[4,5,3,1,2],[2,1,5,4,3],[3,4,1,2,5]],[[1,3,5,2,4],[5,2,4,1,3],[4,1,3,5,2],[3,5,2,4,1],[2,4,1,3,5]]];; AutRep[6]:=[[(),(),()],[(),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2)(3,4),(1,2)(3,4),(1,2)(3,4)],[(1,2)(3,4),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2)(3,4),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2)(3,4)(5,6),(1,2,3)(4,5,6),(1,2,3,4,5,6)],[(1,2,3),(1,2,3),(1,2,3)],[(1,2,3),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3)(4,5),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3)(4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4),(1,2,3,4),(1,2,3,4)],[(1,2,3,4,5),(1,2,3,4,5),(1,2,3,4,5)]]; LatinRep[6]:=[[[5,3,4,6,1,2],[6,1,5,2,3,4],[2,4,1,5,6,3],[3,6,2,1,4,5],[1,5,3,4,2,6],[4,2,6,3,5,1]],[[1,2,3,4,5,6],[2,1,4,3,6,5],[3,4,5,6,1,2],[4,3,6,5,2,1],[5,6,1,2,3,4],[6,5,2,1,4,3]],[[1,2,3,4,5,6],[2,3,1,5,6,4],[3,1,2,6,4,5],[4,5,6,1,2,3],[5,6,4,2,3,1],[6,4,5,3,1,2]],[[1,2,3,4,5,6],[2,3,4,5,6,1],[3,4,5,6,1,2],[4,5,6,1,2,3],[5,6,1,2,3,4],[6,1,2,3,4,5]],[[1,3,2,5,4,6],[4,2,6,1,5,3],[2,1,3,4,6,5],[3,4,5,6,1,2],[5,6,1,2,3,4],[6,5,4,3,2,1]],[[1,4,3,6,5,2],[3,2,5,4,1,6],[2,3,4,5,6,1],[4,5,6,1,2,3],[5,6,1,2,3,4],[6,1,2,3,4,5]],[[1,3,5,6,2,4],[4,2,6,5,1,3],[5,6,1,3,4,2],[6,5,4,2,3,1],[2,1,3,4,5,6],[3,4,2,1,6,5]],[[1,3,2,5,4,6],[4,2,6,1,5,3],[2,4,1,6,3,5],[3,1,5,2,6,4],[5,6,3,4,1,2],[6,5,4,3,2,1]],[[1,4,3,6,5,2],[3,2,5,4,1,6],[2,1,4,3,6,5],[6,3,2,5,4,1],[4,5,6,1,2,3],[5,6,1,2,3,4]],[[1,5,3,2,6,4],[4,2,6,5,3,1],[2,6,4,1,5,3],[5,3,1,4,2,6],[3,1,5,6,4,2],[6,4,2,3,1,5]],[[4,5,6,1,2,3],[6,4,5,2,3,1],[5,6,4,3,1,2],[1,2,3,4,5,6],[2,3,1,5,6,4],[3,1,2,6,4,5]],[[1,3,2,4,6,5],[3,2,1,6,5,4],[2,1,3,5,4,6],[4,5,6,1,2,3],[5,6,4,2,3,1],[6,4,5,3,1,2]],[[1,3,2,4,6,5],[6,2,4,3,5,1],[2,1,3,5,4,6],[3,4,5,6,1,2],[4,5,6,1,2,3],[5,6,1,2,3,4]],[[1,3,2,4,6,5],[6,2,4,3,5,1],[2,1,3,5,4,6],[3,6,5,2,1,4],[5,4,1,6,3,2],[4,5,6,1,2,3]],[[1,3,2,4,6,5],[3,2,1,6,5,4],[2,1,3,5,4,6],[4,6,5,1,3,2],[6,5,4,3,2,1],[5,4,6,2,1,3]],[[1,3,2,4,6,5],[6,2,4,3,5,1],[2,1,3,5,4,6],[3,5,1,6,2,4],[5,4,6,2,1,3],[4,6,5,1,3,2]],[[1,3,5,6,2,4],[6,2,4,5,3,1],[5,6,3,1,4,2],[2,5,6,4,1,3],[3,4,1,2,5,6],[4,1,2,3,6,5]],[[1,3,5,2,6,4],[6,2,4,1,3,5],[4,6,3,5,2,1],[3,5,6,4,1,2],[2,4,1,6,5,3],[5,1,2,3,4,6]]]; AutRep[7]:=[[(),(),()],[(),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2)(3,4),(1,2)(3,4),(1,2)(3,4)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3,4),(1,2,3,4),(1,2,3,4)],[(1,2,3,4)(5,6),(1,2,3,4)(5,6),(1,2,3,4)(5,6)],[(1,2,3,4,5),(1,2,3,4,5),(1,2,3,4,5)],[(1,2,3,4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)]]; LatinRep[7]:=[[[4,7,6,2,3,5,1],[6,1,3,7,5,4,2],[7,2,4,5,6,1,3],[3,5,1,4,2,7,6],[1,4,2,3,7,6,5],[2,6,5,1,4,3,7],[5,3,7,6,1,2,4]],[[1,2,3,4,5,6,7],[2,3,4,5,6,7,1],[3,4,5,6,7,1,2],[4,5,6,7,1,2,3],[5,6,7,1,2,3,4],[6,7,1,2,3,4,5],[7,1,2,3,4,5,6]],[[1,5,6,7,2,3,4],[5,2,7,6,1,4,3],[6,7,3,5,4,1,2],[7,6,5,4,3,2,1],[2,1,4,3,7,6,5],[3,4,1,2,6,5,7],[4,3,2,1,5,7,6]],[[1,3,2,4,5,7,6],[4,2,3,1,7,6,5],[2,4,5,7,6,1,3],[3,1,7,6,2,5,4],[5,7,6,3,1,4,2],[7,6,4,5,3,2,1],[6,5,1,2,4,3,7]],[[1,3,2,4,6,7,5],[3,2,1,7,5,4,6],[2,1,3,5,7,6,4],[4,6,7,1,3,5,2],[7,5,4,6,2,1,3],[5,7,6,2,4,3,1],[6,4,5,3,1,2,7]],[[1,5,6,7,2,3,4],[7,2,5,6,3,4,1],[6,7,3,5,4,1,2],[5,6,7,4,1,2,3],[2,3,4,1,5,6,7],[3,4,1,2,6,7,5],[4,1,2,3,7,5,6]],[[1,5,7,6,2,4,3],[5,2,6,7,1,3,4],[7,6,3,5,4,2,1],[6,7,5,4,3,1,2],[2,1,4,3,5,7,6],[4,3,2,1,7,6,5],[3,4,1,2,6,5,7]],[[1,3,2,6,7,4,5],[7,2,4,3,6,5,1],[6,7,3,5,4,1,2],[5,6,7,4,1,2,3],[2,1,6,7,5,3,4],[3,4,5,1,2,6,7],[4,5,1,2,3,7,6]],[[1,3,2,6,7,4,5],[5,2,4,3,1,7,6],[7,6,3,5,4,2,1],[3,7,1,4,6,5,2],[6,4,7,2,5,1,3],[2,1,5,7,3,6,4],[4,5,6,1,2,3,7]],[[1,3,5,7,2,4,6],[7,2,4,6,1,3,5],[6,1,3,5,7,2,4],[5,7,2,4,6,1,3],[4,6,1,3,5,7,2],[3,5,7,2,4,6,1],[2,4,6,1,3,5,7]]]; AutRep[8]:=[[(),(),()],[(),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2)(3,4),(1,2)(3,4),(1,2)(3,4)],[(1,2)(3,4),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2)(3,4),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3)(4,5,6),(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8)],[(1,2,3,4),(1,2,3,4),(1,2,3,4)],[(1,2,3,4),(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6)(7,8)],[(1,2,3,4),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4)(5,6),(1,2,3,4)(5,6),(1,2,3,4)(5,6)],[(1,2,3,4)(5,6),(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6)(7,8)],[(1,2,3,4)(5,6),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6)(7,8)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5),(1,2,3,4,5),(1,2,3,4,5)],[(1,2,3,4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4,5,6),(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)]]; LatinRep[8]:=[[[2,6,3,4,1,5,8,7],[1,4,7,2,3,6,5,8],[6,2,1,8,5,4,7,3],[5,8,2,7,6,3,4,1],[3,7,5,1,4,8,6,2],[8,5,6,3,7,2,1,4],[4,1,8,5,2,7,3,6],[7,3,4,6,8,1,2,5]],[[1,2,3,4,5,6,7,8],[2,1,4,3,6,5,8,7],[3,4,1,2,7,8,5,6],[4,3,2,1,8,7,6,5],[5,6,7,8,1,2,3,4],[6,5,8,7,2,1,4,3],[7,8,5,6,3,4,1,2],[8,7,6,5,4,3,2,1]],[[1,2,3,4,5,6,7,8],[2,3,4,1,6,7,8,5],[3,4,1,2,7,8,5,6],[4,1,2,3,8,5,6,7],[5,6,7,8,1,2,3,4],[6,7,8,5,2,3,4,1],[7,8,5,6,3,4,1,2],[8,5,6,7,4,1,2,3]],[[1,2,3,4,5,6,7,8],[2,3,4,5,6,7,8,1],[3,4,5,6,7,8,1,2],[4,5,6,7,8,1,2,3],[5,6,7,8,1,2,3,4],[6,7,8,1,2,3,4,5],[7,8,1,2,3,4,5,6],[8,1,2,3,4,5,6,7]],[[1,3,2,4,5,7,6,8],[4,2,3,1,8,6,7,5],[2,1,4,3,6,5,8,7],[3,4,1,2,7,8,5,6],[5,6,7,8,1,2,3,4],[6,5,8,7,2,1,4,3],[7,8,5,6,3,4,1,2],[8,7,6,5,4,3,2,1]],[[1,4,3,2,5,8,7,6],[3,2,1,4,7,6,5,8],[2,3,4,1,6,7,8,5],[4,1,2,3,8,5,6,7],[5,6,7,8,1,2,3,4],[6,7,8,5,2,3,4,1],[7,8,5,6,3,4,1,2],[8,5,6,7,4,1,2,3]],[[1,4,3,6,5,8,7,2],[3,2,5,4,7,6,1,8],[2,3,4,5,6,7,8,1],[4,5,6,7,8,1,2,3],[5,6,7,8,1,2,3,4],[6,7,8,1,2,3,4,5],[7,8,1,2,3,4,5,6],[8,1,2,3,4,5,6,7]],[[5,6,7,8,1,2,3,4],[6,5,8,7,2,1,4,3],[7,8,5,6,3,4,1,2],[8,7,6,5,4,3,2,1],[1,2,3,4,7,8,5,6],[2,1,4,3,5,6,7,8],[3,4,1,2,8,7,6,5],[4,3,2,1,6,5,8,7]],[[1,3,2,4,5,7,6,8],[4,2,3,1,8,6,7,5],[2,4,1,3,6,8,5,7],[3,1,4,2,7,5,8,6],[5,6,7,8,1,2,3,4],[6,5,8,7,2,1,4,3],[7,8,5,6,3,4,1,2],[8,7,6,5,4,3,2,1]],[[1,4,3,2,5,8,7,6],[3,2,1,4,7,6,5,8],[2,1,4,3,6,5,8,7],[4,3,2,1,8,7,6,5],[5,6,7,8,1,2,3,4],[6,7,8,5,2,3,4,1],[7,8,5,6,3,4,1,2],[8,5,6,7,4,1,2,3]],[[1,4,3,6,5,8,7,2],[3,2,5,4,7,6,1,8],[2,1,4,3,6,5,8,7],[8,3,2,5,4,7,6,1],[4,5,6,7,8,1,2,3],[5,6,7,8,1,2,3,4],[6,7,8,1,2,3,4,5],[7,8,1,2,3,4,5,6]],[[1,3,2,4,7,8,5,6],[4,2,3,1,8,7,6,5],[2,5,7,8,1,6,3,4],[6,1,8,7,5,2,4,3],[7,8,4,5,6,3,1,2],[8,7,6,3,4,5,2,1],[3,4,5,6,2,1,8,7],[5,6,1,2,3,4,7,8]],[[1,3,2,4,5,7,6,8],[4,2,3,1,8,6,7,5],[2,4,1,3,6,8,5,7],[3,1,4,2,7,5,8,6],[5,7,6,8,1,3,2,4],[8,6,7,5,4,2,3,1],[6,5,8,7,2,1,4,3],[7,8,5,6,3,4,1,2]],[[1,4,3,2,5,8,7,6],[3,2,1,4,7,6,5,8],[2,1,4,3,6,5,8,7],[4,3,2,1,8,7,6,5],[5,8,7,6,1,4,3,2],[7,6,5,8,3,2,1,4],[6,7,8,5,2,3,4,1],[8,5,6,7,4,1,2,3]],[[1,4,3,6,5,8,7,2],[3,2,5,4,7,6,1,8],[2,1,4,3,6,5,8,7],[8,3,2,5,4,7,6,1],[4,7,6,1,8,3,2,5],[6,5,8,7,2,1,4,3],[5,6,7,8,1,2,3,4],[7,8,1,2,3,4,5,6]],[[1,3,2,4,5,7,6,8],[4,2,3,1,8,6,7,5],[2,4,1,3,6,8,5,7],[3,1,4,2,7,5,8,6],[5,7,6,8,1,3,2,4],[8,6,7,5,4,2,3,1],[6,8,5,7,2,4,1,3],[7,5,8,6,3,1,4,2]],[[1,4,3,2,5,8,7,6],[3,2,1,4,7,6,5,8],[2,1,4,3,6,5,8,7],[4,3,2,1,8,7,6,5],[5,8,7,6,1,4,3,2],[7,6,5,8,3,2,1,4],[6,5,8,7,2,1,4,3],[8,7,6,5,4,3,2,1]],[[1,4,3,6,5,8,7,2],[3,2,5,4,7,6,1,8],[2,1,4,3,6,5,8,7],[8,3,2,5,4,7,6,1],[4,7,6,1,8,3,2,5],[6,5,8,7,2,1,4,3],[5,8,7,2,1,4,3,6],[7,6,1,8,3,2,5,4]],[[1,3,2,4,7,8,5,6],[3,2,1,8,5,7,6,4],[2,1,3,7,8,6,4,5],[4,7,8,1,6,5,2,3],[8,5,7,6,2,4,3,1],[7,8,6,5,4,3,1,2],[5,6,4,2,3,1,7,8],[6,4,5,3,1,2,8,7]],[[1,3,7,4,6,8,2,5],[7,2,4,8,5,1,6,3],[2,8,3,5,7,6,4,1],[3,1,8,6,4,7,5,2],[8,4,2,7,1,5,3,6],[6,7,5,3,8,2,1,4],[4,5,6,1,2,3,7,8],[5,6,1,2,3,4,8,7]],[[5,6,7,8,1,2,3,4],[8,5,6,7,2,3,4,1],[7,8,5,6,3,4,1,2],[6,7,8,5,4,1,2,3],[1,2,3,4,6,7,8,5],[2,3,4,1,7,8,5,6],[3,4,1,2,8,5,6,7],[4,1,2,3,5,6,7,8]],[[5,7,6,8,1,3,2,4],[7,6,8,5,4,2,1,3],[6,8,5,7,3,1,4,2],[8,5,7,6,2,4,3,1],[1,2,3,4,6,5,7,8],[4,1,2,3,5,6,8,7],[3,4,1,2,7,8,5,6],[2,3,4,1,8,7,6,5]],[[1,3,2,5,4,6,8,7],[6,2,4,3,8,1,7,5],[4,7,3,1,6,5,2,8],[2,1,8,4,5,7,6,3],[3,4,1,2,7,8,5,6],[5,6,7,8,1,2,3,4],[7,8,5,6,2,3,4,1],[8,5,6,7,3,4,1,2]],[[1,3,2,8,5,7,6,4],[5,2,4,3,1,6,8,7],[8,6,3,5,4,2,7,1],[2,1,7,4,6,5,3,8],[3,4,5,6,7,8,1,2],[4,5,6,7,8,1,2,3],[6,7,8,1,2,3,4,5],[7,8,1,2,3,4,5,6]],[[5,7,6,8,1,3,2,4],[8,6,7,5,4,2,3,1],[6,8,5,7,3,1,4,2],[7,5,8,6,2,4,1,3],[1,4,3,2,8,7,5,6],[3,2,1,4,7,8,6,5],[2,3,4,1,5,6,7,8],[4,1,2,3,6,5,8,7]],[[5,7,6,8,1,3,2,4],[7,6,8,5,4,2,1,3],[6,8,5,7,3,1,4,2],[8,5,7,6,2,4,3,1],[1,4,3,2,7,5,6,8],[3,2,1,4,6,8,7,5],[2,3,4,1,5,6,8,7],[4,1,2,3,8,7,5,6]],[[1,3,2,5,4,6,8,7],[6,2,4,3,8,1,7,5],[4,7,3,1,6,5,2,8],[2,1,8,4,5,7,6,3],[3,5,1,7,2,8,4,6],[8,4,6,2,7,3,5,1],[5,6,7,8,1,2,3,4],[7,8,5,6,3,4,1,2]],[[1,3,2,8,5,7,6,4],[5,2,4,3,1,6,8,7],[8,6,3,5,4,2,7,1],[2,1,7,4,6,5,3,8],[3,8,5,2,7,4,1,6],[7,4,1,6,3,8,5,2],[4,5,6,7,8,1,2,3],[6,7,8,1,2,3,4,5]],[[5,7,6,8,1,3,2,4],[7,6,8,5,4,2,1,3],[6,8,5,7,3,1,4,2],[8,5,7,6,2,4,3,1],[1,4,3,2,6,8,5,7],[3,2,1,4,7,5,8,6],[2,1,4,3,5,7,6,8],[4,3,2,1,8,6,7,5]],[[1,3,2,5,4,6,8,7],[6,2,4,3,8,1,7,5],[4,7,3,1,6,5,2,8],[2,1,8,4,5,7,6,3],[3,5,1,7,2,8,4,6],[8,4,6,2,7,3,5,1],[5,8,7,6,1,4,3,2],[7,6,5,8,3,2,1,4]],[[1,3,2,8,5,7,6,4],[5,2,4,3,1,6,8,7],[8,6,3,5,4,2,7,1],[2,1,7,4,6,5,3,8],[3,8,5,2,7,4,1,6],[7,4,1,6,3,8,5,2],[4,7,6,1,8,3,2,5],[6,5,8,7,2,1,4,3]],[[1,3,2,5,4,6,8,7],[6,2,4,3,8,1,7,5],[4,7,3,1,6,5,2,8],[2,1,8,4,5,7,6,3],[3,5,7,6,1,8,4,2],[7,4,6,8,3,2,5,1],[5,8,1,7,2,4,3,6],[8,6,5,2,7,3,1,4]],[[1,3,2,8,5,7,6,4],[5,2,4,3,1,6,8,7],[8,6,3,5,4,2,7,1],[2,1,7,4,6,5,3,8],[3,5,8,2,7,1,4,6],[7,4,6,1,3,8,2,5],[6,8,5,7,2,4,1,3],[4,7,1,6,8,3,5,2]],[[1,3,6,7,8,2,4,5],[8,2,4,6,7,3,5,1],[7,8,3,5,6,4,1,2],[6,7,8,4,1,5,2,3],[2,6,7,8,5,1,3,4],[3,4,5,1,2,6,7,8],[4,5,1,2,3,7,8,6],[5,1,2,3,4,8,6,7]],[[1,3,2,6,7,8,4,5],[8,2,4,3,1,7,5,6],[7,8,3,5,4,2,6,1],[3,7,8,4,6,5,1,2],[6,4,7,8,5,1,2,3],[2,1,5,7,8,6,3,4],[4,5,6,1,2,3,7,8],[5,6,1,2,3,4,8,7]],[[1,3,5,7,2,8,4,6],[7,2,4,6,8,3,1,5],[4,8,3,5,1,7,6,2],[8,5,7,4,6,2,3,1],[3,7,6,8,5,1,2,4],[2,4,8,1,7,6,5,3],[5,6,1,2,3,4,7,8],[6,1,2,3,4,5,8,7]],[[1,3,5,7,2,4,8,6],[8,2,4,6,1,3,5,7],[6,8,3,5,7,2,4,1],[5,7,8,4,6,1,3,2],[4,6,1,8,5,7,2,3],[3,5,7,2,8,6,1,4],[2,4,6,1,3,8,7,5],[7,1,2,3,4,5,6,8]]]; AutRep[9]:=[[(),(),()],[(),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2,3),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3)(4,5)(6,7)(8,9),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4,5),(1,2,3,4,5),(1,2,3,4,5)],[(1,2,3,4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4,5,6),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)]]; LatinRep[9]:=[[[5,8,7,9,6,3,4,1,2],[9,5,2,7,3,1,6,4,8],[8,4,6,3,7,5,9,2,1],[7,6,3,1,2,8,5,9,4],[1,2,4,5,9,7,8,6,3],[3,9,1,4,5,2,7,8,6],[4,1,9,2,8,6,3,7,5],[2,3,8,6,4,9,1,5,7],[6,7,5,8,1,4,2,3,9]],[[1,2,3,4,5,6,7,8,9],[2,3,1,5,6,4,8,9,7],[3,1,2,6,4,5,9,7,8],[4,5,6,7,8,9,1,2,3],[5,6,4,8,9,7,2,3,1],[6,4,5,9,7,8,3,1,2],[7,8,9,1,2,3,4,5,6],[8,9,7,2,3,1,5,6,4],[9,7,8,3,1,2,6,4,5]],[[1,2,3,4,5,6,7,8,9],[2,3,4,5,6,7,8,9,1],[3,4,5,6,7,8,9,1,2],[4,5,6,7,8,9,1,2,3],[5,6,7,8,9,1,2,3,4],[6,7,8,9,1,2,3,4,5],[7,8,9,1,2,3,4,5,6],[8,9,1,2,3,4,5,6,7],[9,1,2,3,4,5,6,7,8]],[[1,3,2,7,8,9,4,5,6],[4,2,7,1,9,8,3,6,5],[2,7,8,9,3,5,6,1,4],[7,1,9,8,6,4,5,2,3],[8,9,3,5,7,6,1,4,2],[9,8,6,4,5,7,2,3,1],[3,4,5,6,1,2,7,9,8],[5,6,1,2,4,3,9,8,7],[6,5,4,3,2,1,8,7,9]],[[1,7,3,9,5,8,2,6,4],[9,2,8,4,7,6,5,3,1],[2,8,4,7,6,9,1,5,3],[7,3,9,5,8,1,4,2,6],[3,9,5,8,1,7,6,4,2],[8,4,7,6,9,2,3,1,5],[4,5,6,1,2,3,7,8,9],[5,6,1,2,3,4,8,9,7],[6,1,2,3,4,5,9,7,8]],[[1,3,2,4,5,7,6,9,8],[4,2,3,1,8,6,9,5,7],[2,4,1,3,7,9,5,8,6],[3,1,4,2,9,8,7,6,5],[5,7,6,9,1,3,8,2,4],[8,6,9,5,4,2,1,7,3],[7,9,5,8,6,4,2,3,1],[9,8,7,6,3,5,4,1,2],[6,5,8,7,2,1,3,4,9]],[[1,3,2,4,6,5,7,9,8],[3,2,1,6,5,4,9,8,7],[2,1,3,5,4,6,8,7,9],[4,5,6,7,8,9,1,2,3],[5,6,4,8,9,7,2,3,1],[6,4,5,9,7,8,3,1,2],[7,8,9,1,2,3,4,5,6],[8,9,7,2,3,1,5,6,4],[9,7,8,3,1,2,6,4,5]],[[1,3,2,4,6,5,7,9,8],[9,2,4,3,5,7,6,8,1],[2,1,3,5,4,6,8,7,9],[3,4,5,6,7,8,9,1,2],[4,5,6,7,8,9,1,2,3],[5,6,7,8,9,1,2,3,4],[6,7,8,9,1,2,3,4,5],[7,8,9,1,2,3,4,5,6],[8,9,1,2,3,4,5,6,7]],[[1,3,2,4,6,5,7,9,8],[6,2,4,3,5,1,9,8,7],[2,1,3,5,4,6,8,7,9],[7,4,9,6,8,2,1,5,3],[3,8,5,7,1,9,4,2,6],[4,9,6,8,2,7,3,1,5],[8,5,7,1,9,3,6,4,2],[5,7,1,9,3,8,2,6,4],[9,6,8,2,7,4,5,3,1]],[[1,3,2,7,8,9,4,5,6],[3,2,1,9,7,8,5,6,4],[2,1,3,8,9,7,6,4,5],[7,8,9,4,6,5,1,2,3],[9,7,8,6,5,4,2,3,1],[8,9,7,5,4,6,3,1,2],[4,5,6,1,2,3,9,8,7],[5,6,4,2,3,1,8,7,9],[6,4,5,3,1,2,7,9,8]],[[1,3,2,4,6,5,7,9,8],[3,2,1,6,5,4,9,8,7],[2,1,3,5,4,6,8,7,9],[4,6,5,7,9,8,1,3,2],[6,5,4,9,8,7,3,2,1],[5,4,6,8,7,9,2,1,3],[7,8,9,1,2,3,4,5,6],[8,9,7,2,3,1,5,6,4],[9,7,8,3,1,2,6,4,5]],[[1,3,2,4,6,5,7,9,8],[9,2,4,3,5,7,6,8,1],[2,1,3,5,4,6,8,7,9],[3,5,1,6,8,4,9,2,7],[8,4,6,2,7,9,5,1,3],[4,9,5,7,3,8,1,6,2],[5,6,7,8,9,1,2,3,4],[6,7,8,9,1,2,3,4,5],[7,8,9,1,2,3,4,5,6]],[[1,3,2,4,6,5,7,9,8],[3,2,1,6,5,4,9,8,7],[2,1,3,5,4,6,8,7,9],[4,6,5,7,9,8,1,3,2],[6,5,4,9,8,7,3,2,1],[5,4,6,8,7,9,2,1,3],[7,9,8,1,3,2,4,6,5],[9,8,7,3,2,1,6,5,4],[8,7,9,2,1,3,5,4,6]],[[1,3,2,4,6,5,7,9,8],[9,2,4,3,5,7,6,8,1],[2,1,3,5,4,6,8,7,9],[3,5,1,6,8,4,9,2,7],[8,4,6,2,7,9,5,1,3],[4,9,5,7,3,8,1,6,2],[5,7,9,8,1,3,2,4,6],[7,6,8,1,9,2,4,3,5],[6,8,7,9,2,1,3,5,4]],[[1,3,2,5,4,6,8,9,7],[6,2,4,3,9,1,7,5,8],[4,7,3,1,6,9,2,8,5],[2,1,8,4,5,7,9,3,6],[3,5,7,9,1,8,6,2,4],[9,4,6,8,3,2,5,7,1],[5,9,1,7,8,4,3,6,2],[8,6,9,2,7,5,1,4,3],[7,8,5,6,2,3,4,1,9]],[[1,6,7,8,9,2,3,4,5],[9,2,6,7,8,3,4,5,1],[8,9,3,6,7,4,5,1,2],[7,8,9,4,6,5,1,2,3],[6,7,8,9,5,1,2,3,4],[2,3,4,5,1,7,8,9,6],[3,4,5,1,2,8,9,6,7],[4,5,1,2,3,9,6,7,8],[5,1,2,3,4,6,7,8,9]],[[1,3,2,7,8,9,4,5,6],[9,2,4,3,7,8,5,6,1],[8,9,3,5,4,7,6,1,2],[7,8,9,4,6,5,1,2,3],[6,7,8,9,5,1,2,3,4],[2,1,7,8,9,6,3,4,5],[3,4,5,6,1,2,7,8,9],[4,5,6,1,2,3,8,9,7],[5,6,1,2,3,4,9,7,8]],[[1,3,2,7,9,8,4,6,5],[9,2,4,3,8,7,6,5,1],[8,7,3,5,4,9,2,1,6],[7,9,8,4,6,5,1,3,2],[6,8,7,9,5,1,3,2,4],[2,1,9,8,7,6,5,4,3],[3,4,5,6,1,2,7,8,9],[4,5,6,1,2,3,8,9,7],[5,6,1,2,3,4,9,7,8]],[[1,3,2,7,9,8,4,6,5],[7,2,4,3,8,9,1,5,6],[9,8,3,5,4,7,6,2,1],[8,9,7,4,6,5,3,1,2],[6,7,9,8,5,1,2,4,3],[2,1,8,9,7,6,5,3,4],[3,6,5,2,1,4,7,9,8],[5,4,1,6,3,2,9,8,7],[4,5,6,1,2,3,8,7,9]],[[1,3,2,7,9,8,4,6,5],[9,2,4,3,8,7,6,5,1],[8,7,3,5,4,9,2,1,6],[7,9,8,4,6,5,1,3,2],[6,8,7,9,5,1,3,2,4],[2,1,9,8,7,6,5,4,3],[3,5,1,6,2,4,7,9,8],[5,4,6,2,1,3,9,8,7],[4,6,5,1,3,2,8,7,9]],[[1,3,2,6,8,4,9,5,7],[9,2,4,3,7,8,5,6,1],[6,9,3,5,4,1,8,7,2],[8,7,9,4,6,5,2,1,3],[3,8,1,9,5,7,6,2,4],[7,4,8,2,9,6,1,3,5],[2,1,5,8,3,9,7,4,6],[4,5,6,7,1,2,3,8,9],[5,6,7,1,2,3,4,9,8]],[[1,3,2,6,8,4,9,5,7],[6,2,4,3,7,1,5,9,8],[9,7,3,5,4,8,2,6,1],[7,9,8,4,6,5,1,3,2],[4,8,9,1,5,7,6,2,3],[3,5,1,9,2,6,8,7,4],[8,4,6,2,9,3,7,1,5],[2,1,5,7,3,9,4,8,6],[5,6,7,8,1,2,3,4,9]],[[1,3,2,7,9,8,4,6,5],[6,2,4,3,8,1,9,5,7],[8,7,3,5,4,9,2,1,6],[7,9,8,4,6,5,1,3,2],[3,8,1,9,5,7,6,2,4],[5,4,9,2,1,6,8,7,3],[4,6,5,1,3,2,7,9,8],[9,5,7,6,2,4,3,8,1],[2,1,6,8,7,3,5,4,9]]]; AutRep[10]:=[[(),(),()],[(),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2)(3,4),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3)(4,5,6),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3)(4,5,6)(7,8),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3)(4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10)],[(1,2,3,4,5),(1,2,3,4,5),(1,2,3,4,5)],[(1,2,3,4,5),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5)(6,7),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5)(6,7)(8,9),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4,5,6),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8)],[(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)]]; LatinRep[10]:=[[[4,1,9,5,2,8,10,3,7,6],[5,3,10,9,8,7,2,1,6,4],[6,7,5,10,9,1,4,8,3,2],[3,9,7,1,4,10,8,6,2,5],[1,6,8,2,3,4,5,7,9,10],[10,4,2,3,5,6,7,9,1,8],[9,2,1,8,7,5,6,4,10,3],[7,8,6,4,10,3,9,2,5,1],[8,5,3,7,6,2,1,10,4,9],[2,10,4,6,1,9,3,5,8,7]],[[1,2,3,4,5,6,7,8,9,10],[2,1,4,3,6,5,8,7,10,9],[3,4,1,2,7,8,9,10,5,6],[4,3,2,1,8,7,10,9,6,5],[5,6,7,8,9,10,1,2,3,4],[6,5,8,7,10,9,2,1,4,3],[7,8,9,10,3,4,5,6,1,2],[8,7,10,9,4,3,6,5,2,1],[9,10,5,6,1,2,3,4,7,8],[10,9,6,5,2,1,4,3,8,7]],[[1,2,3,4,5,6,7,8,9,10],[2,3,4,5,1,7,8,9,10,6],[3,4,5,1,2,8,9,10,6,7],[4,5,1,2,3,9,10,6,7,8],[5,1,2,3,4,10,6,7,8,9],[6,7,8,9,10,1,2,3,4,5],[7,8,9,10,6,2,3,4,5,1],[8,9,10,6,7,3,4,5,1,2],[9,10,6,7,8,4,5,1,2,3],[10,6,7,8,9,5,1,2,3,4]],[[1,2,3,4,5,6,7,8,9,10],[2,3,4,5,6,7,8,9,10,1],[3,4,5,6,7,8,9,10,1,2],[4,5,6,7,8,9,10,1,2,3],[5,6,7,8,9,10,1,2,3,4],[6,7,8,9,10,1,2,3,4,5],[7,8,9,10,1,2,3,4,5,6],[8,9,10,1,2,3,4,5,6,7],[9,10,1,2,3,4,5,6,7,8],[10,1,2,3,4,5,6,7,8,9]],[[1,3,2,4,5,7,6,9,8,10],[4,2,3,1,8,6,10,5,9,7],[2,1,4,3,6,5,7,8,10,9],[3,4,1,2,7,8,9,10,5,6],[5,6,7,8,9,10,1,2,3,4],[6,5,8,7,10,9,2,1,4,3],[7,8,9,10,1,2,3,4,6,5],[8,7,10,9,3,4,5,6,1,2],[9,10,5,6,2,1,4,3,7,8],[10,9,6,5,4,3,8,7,2,1]],[[1,4,3,6,5,8,7,10,9,2],[3,2,5,4,7,6,9,8,1,10],[2,3,4,5,6,7,8,9,10,1],[4,5,6,7,8,9,10,1,2,3],[5,6,7,8,9,10,1,2,3,4],[6,7,8,9,10,1,2,3,4,5],[7,8,9,10,1,2,3,4,5,6],[8,9,10,1,2,3,4,5,6,7],[9,10,1,2,3,4,5,6,7,8],[10,1,2,3,4,5,6,7,8,9]],[[1,3,2,4,5,7,6,9,8,10],[4,2,3,1,8,6,10,5,9,7],[2,4,1,3,6,8,5,10,7,9],[3,1,4,2,7,5,9,6,10,8],[5,6,7,8,9,10,1,2,3,4],[6,5,8,7,10,9,2,1,4,3],[7,8,9,10,1,2,3,4,5,6],[8,7,10,9,2,1,4,3,6,5],[9,10,5,6,3,4,7,8,1,2],[10,9,6,5,4,3,8,7,2,1]],[[1,4,3,6,5,8,7,10,9,2],[3,2,5,4,7,6,9,8,1,10],[2,1,4,3,6,5,8,7,10,9],[10,3,2,5,4,7,6,9,8,1],[4,5,6,7,8,9,10,1,2,3],[5,6,7,8,9,10,1,2,3,4],[6,7,8,9,10,1,2,3,4,5],[7,8,9,10,1,2,3,4,5,6],[8,9,10,1,2,3,4,5,6,7],[9,10,1,2,3,4,5,6,7,8]],[[10,6,7,8,9,1,3,5,2,4],[5,10,8,7,2,9,4,6,1,3],[3,9,10,1,7,8,5,2,4,6],[9,4,2,10,8,7,6,1,3,5],[8,7,5,9,10,3,2,4,6,1],[7,8,9,6,4,10,1,3,5,2],[2,1,4,3,6,5,9,10,7,8],[4,3,6,5,1,2,10,7,8,9],[6,5,1,2,3,4,7,8,9,10],[1,2,3,4,5,6,8,9,10,7]],[[1,3,2,4,5,7,6,9,8,10],[4,2,3,1,8,6,10,5,9,7],[2,4,1,3,6,8,5,10,7,9],[3,1,4,2,7,5,9,6,10,8],[5,7,6,9,1,10,3,8,2,4],[8,6,10,5,9,2,7,4,3,1],[6,5,7,8,10,9,1,2,4,3],[7,8,9,10,2,1,4,3,5,6],[9,10,5,6,3,4,8,7,1,2],[10,9,8,7,4,3,2,1,6,5]],[[1,4,3,6,5,8,7,10,9,2],[3,2,5,4,7,6,9,8,1,10],[2,1,4,3,6,5,8,7,10,9],[10,3,2,5,4,7,6,9,8,1],[4,7,6,9,8,1,10,3,2,5],[6,5,8,7,10,9,2,1,4,3],[5,6,7,8,9,10,1,2,3,4],[7,8,9,10,1,2,3,4,5,6],[8,9,10,1,2,3,4,5,6,7],[9,10,1,2,3,4,5,6,7,8]],[[1,3,2,4,5,7,9,10,6,8],[4,2,3,1,8,6,10,9,5,7],[2,4,1,3,9,10,5,7,8,6],[3,1,4,2,10,9,8,6,7,5],[5,7,9,10,1,3,6,8,2,4],[8,6,10,9,4,2,7,5,1,3],[9,10,5,7,6,8,1,3,4,2],[10,9,8,6,7,5,4,2,3,1],[6,5,7,8,2,1,3,4,9,10],[7,8,6,5,3,4,2,1,10,9]],[[1,3,2,4,5,7,6,9,8,10],[4,2,3,1,8,6,10,5,9,7],[2,4,1,3,6,8,5,10,7,9],[3,1,4,2,7,5,9,6,10,8],[5,7,6,9,1,10,3,8,2,4],[8,6,10,5,9,2,7,4,3,1],[6,8,5,10,2,9,4,7,1,3],[7,5,9,6,10,1,8,3,4,2],[9,10,7,8,3,4,1,2,5,6],[10,9,8,7,4,3,2,1,6,5]],[[1,4,3,6,5,8,7,10,9,2],[3,2,5,4,7,6,9,8,1,10],[2,1,4,3,6,5,8,7,10,9],[10,3,2,5,4,7,6,9,8,1],[4,7,6,9,8,1,10,3,2,5],[6,5,8,7,10,9,2,1,4,3],[5,8,7,10,9,2,1,4,3,6],[7,6,9,8,1,10,3,2,5,4],[8,9,10,1,2,3,4,5,6,7],[9,10,1,2,3,4,5,6,7,8]],[[1,7,3,9,5,2,8,4,10,6],[6,2,8,4,10,7,3,9,5,1],[2,8,4,10,6,1,7,3,9,5],[7,3,9,5,1,6,2,8,4,10],[3,9,5,1,7,4,10,6,2,8],[8,4,10,6,2,9,5,1,7,3],[4,10,6,2,8,3,9,5,1,7],[9,5,1,7,3,8,4,10,6,2],[5,1,7,3,9,10,6,2,8,4],[10,6,2,8,4,5,1,7,3,9]],[[10,9,8,7,6,1,4,2,5,3],[8,10,9,2,7,4,5,3,6,1],[9,8,10,5,3,7,6,1,4,2],[4,7,1,10,9,8,2,5,3,6],[2,5,7,8,10,9,3,6,1,4],[7,3,6,9,8,10,1,4,2,5],[5,6,4,3,1,2,9,10,7,8],[3,1,2,6,4,5,10,7,8,9],[6,4,5,1,2,3,7,8,9,10],[1,2,3,4,5,6,8,9,10,7]],[[1,7,9,4,8,10,2,5,3,6],[9,2,8,10,5,7,6,3,1,4],[8,10,3,7,9,6,4,1,5,2],[2,8,10,5,7,9,1,4,6,3],[10,3,7,9,6,8,5,2,4,1],[7,9,4,8,10,1,3,6,2,5],[3,4,5,6,1,2,8,7,9,10],[6,1,2,3,4,5,7,8,10,9],[5,6,1,2,3,4,9,10,7,8],[4,5,6,1,2,3,10,9,8,7]],[[1,7,9,4,8,10,2,5,3,6],[9,2,8,10,5,7,6,3,1,4],[8,10,3,7,9,6,4,1,5,2],[2,8,10,5,7,9,1,4,6,3],[10,3,7,9,6,8,5,2,4,1],[7,9,4,8,10,1,3,6,2,5],[3,6,5,2,1,4,9,7,8,10],[5,4,1,6,3,2,8,10,9,7],[4,5,6,1,2,3,7,8,10,9],[6,1,2,3,4,5,10,9,7,8]],[[1,7,9,4,8,10,2,5,3,6],[9,2,8,10,5,7,6,3,1,4],[8,10,3,7,9,6,4,1,5,2],[2,8,10,5,7,9,1,4,6,3],[10,3,7,9,6,8,5,2,4,1],[7,9,4,8,10,1,3,6,2,5],[3,6,5,2,1,4,8,10,7,9],[5,4,1,6,3,2,9,7,10,8],[4,1,6,3,2,5,7,9,8,10],[6,5,2,1,4,3,10,8,9,7]],[[1,3,2,4,6,5,7,9,10,8],[3,2,1,6,5,4,10,8,7,9],[2,1,3,5,4,6,8,10,9,7],[4,6,5,7,9,10,1,3,8,2],[6,5,4,10,8,7,9,2,1,3],[5,4,6,8,10,9,2,7,3,1],[7,9,10,1,3,8,4,6,2,5],[10,8,7,9,2,1,3,5,4,6],[8,10,9,2,7,3,5,1,6,4],[9,7,8,3,1,2,6,4,5,10]],[[1,3,2,5,4,6,9,10,7,8],[6,2,4,3,10,1,7,9,8,5],[4,7,3,1,9,10,2,8,5,6],[2,1,8,4,5,9,10,3,6,7],[3,5,9,10,6,8,1,7,2,4],[10,4,6,9,8,7,5,2,3,1],[9,10,1,7,3,5,8,6,4,2],[8,9,10,2,7,4,6,5,1,3],[5,6,7,8,1,2,3,4,9,10],[7,8,5,6,2,3,4,1,10,9]],[[1,3,2,5,4,9,7,10,6,8],[6,2,4,3,9,1,10,8,5,7],[4,7,3,1,5,10,2,9,8,6],[2,1,8,4,10,6,9,3,7,5],[3,9,5,10,6,8,1,7,2,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AutRep[11]:=[[(),(),()],[(),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)]]; LatinRep[11]:=[[[5,1,4,9,8,6,3,7,10,2,11],[11,2,3,1,7,10,9,5,4,8,6],[7,6,9,5,11,2,8,10,3,4,1],[4,9,6,10,2,7,1,11,8,3,5],[3,4,10,2,5,1,11,9,7,6,8],[8,11,1,6,9,3,10,4,5,7,2],[6,10,5,11,3,9,4,8,2,1,7],[9,3,8,7,10,11,2,6,1,5,4],[2,7,11,8,6,4,5,1,9,10,3],[1,5,7,3,4,8,6,2,11,9,10],[10,8,2,4,1,5,7,3,6,11,9]],[[1,2,3,4,5,6,7,8,9,10,11],[2,3,4,5,6,7,8,9,10,11,1],[3,4,5,6,7,8,9,10,11,1,2],[4,5,6,7,8,9,10,11,1,2,3],[5,6,7,8,9,10,11,1,2,3,4],[6,7,8,9,10,11,1,2,3,4,5],[7,8,9,10,11,1,2,3,4,5,6],[8,9,10,11,1,2,3,4,5,6,7],[9,10,11,1,2,3,4,5,6,7,8],[10,11,1,2,3,4,5,6,7,8,9],[11,1,2,3,4,5,6,7,8,9,10]],[[11,7,8,9,10,1,3,5,2,4,6],[7,11,9,8,2,10,4,6,1,3,5],[3,10,11,7,8,9,5,2,4,6,1],[10,4,7,11,9,8,6,1,3,5,2],[9,8,5,10,11,7,2,4,6,1,3],[8,9,10,6,7,11,1,3,5,2,4],[5,6,2,1,4,3,9,10,11,7,8],[2,1,4,3,6,5,10,11,7,8,9],[4,3,6,5,1,2,11,7,8,9,10],[6,5,1,2,3,4,7,8,9,10,11],[1,2,3,4,5,6,8,9,10,11,7]],[[11,9,4,8,10,7,6,1,2,3,5],[9,11,7,3,8,10,2,5,1,4,6],[3,8,11,9,6,1,10,2,4,5,7],[7,4,9,11,2,5,1,10,3,6,8],[4,10,5,1,11,9,8,3,6,7,2],[10,3,2,6,9,11,4,7,5,8,1],[5,1,6,10,7,3,11,9,8,2,4],[2,6,10,5,4,8,9,11,7,1,3],[6,5,8,7,1,2,3,4,11,9,10],[8,7,1,2,3,4,5,6,9,10,11],[1,2,3,4,5,6,7,8,10,11,9]],[[11,8,4,7,3,10,6,9,5,1,2],[7,11,8,3,9,4,10,5,2,6,1],[3,7,11,10,6,9,5,1,8,2,4],[8,4,9,11,10,5,2,6,1,7,3],[4,10,5,9,11,1,8,2,7,3,6],[9,3,10,6,2,11,1,7,4,8,5],[5,9,6,1,7,2,11,3,10,4,8],[10,6,2,5,1,8,4,11,3,9,7],[6,1,7,2,8,3,9,4,11,5,10],[2,5,1,8,4,7,3,10,6,11,9],[1,2,3,4,5,6,7,8,9,10,11]],[[11,10,9,8,7,1,4,2,5,3,6],[9,11,10,2,8,7,5,3,6,1,4],[10,9,11,7,3,8,6,1,4,2,5],[4,8,7,11,10,9,2,5,3,6,1],[7,5,8,9,11,10,3,6,1,4,2],[8,7,6,10,9,11,1,4,2,5,3],[2,3,1,5,6,4,9,10,11,7,8],[5,6,4,3,1,2,10,11,7,8,9],[3,1,2,6,4,5,11,7,8,9,10],[6,4,5,1,2,3,7,8,9,10,11],[1,2,3,4,5,6,8,9,10,11,7]],[[8,10,9,11,6,5,3,2,1,4,7],[7,9,10,6,11,4,2,1,3,5,8],[10,8,7,5,4,11,1,3,2,6,9],[4,6,5,3,10,1,11,9,8,7,2],[6,5,4,2,1,10,9,11,7,8,3],[5,4,6,10,3,2,8,7,11,9,1],[3,11,1,7,9,8,6,10,4,2,5],[2,1,11,9,8,7,5,4,10,3,6],[11,3,2,8,7,9,10,6,5,1,4],[9,7,8,1,2,3,4,5,6,10,11],[1,2,3,4,5,6,7,8,9,11,10]],[[11,10,9,6,7,4,8,1,3,5,2],[7,11,10,9,2,8,1,5,4,6,3],[9,8,11,10,6,3,5,2,1,7,4],[10,9,5,11,3,7,4,6,2,8,1],[5,4,8,1,11,10,9,3,7,2,6],[2,6,1,5,4,11,10,9,8,3,7],[6,3,7,2,9,1,11,10,5,4,8],[3,7,4,8,10,9,2,11,6,1,5],[4,1,2,3,8,5,6,7,11,9,10],[8,5,6,7,1,2,3,4,9,10,11],[1,2,3,4,5,6,7,8,10,11,9]],[[8,10,5,9,3,6,2,11,7,1,4],[10,5,9,6,11,4,7,3,2,8,1],[7,9,6,10,4,11,1,8,5,3,2],[9,8,10,7,5,1,11,2,4,6,3],[6,2,7,11,1,10,4,9,3,5,8],[11,7,3,8,10,2,9,1,6,4,5],[5,11,8,4,2,9,3,10,1,7,6],[1,6,11,5,9,3,10,4,8,2,7],[4,3,2,1,8,7,6,5,11,9,10],[2,1,4,3,6,5,8,7,10,11,9],[3,4,1,2,7,8,5,6,9,10,11]],[[11,6,2,7,3,4,9,5,10,1,8],[4,11,7,3,8,2,5,10,1,6,9],[9,5,11,8,4,7,3,1,6,2,10],[5,10,1,11,9,3,8,4,2,7,6],[10,1,6,2,11,8,4,9,5,3,7],[6,9,5,10,1,11,2,7,3,8,4],[2,7,10,1,6,9,11,3,8,4,5],[7,3,8,6,2,5,10,11,4,9,1],[3,8,4,9,7,10,1,6,11,5,2],[8,4,9,5,10,1,6,2,7,11,3],[1,2,3,4,5,6,7,8,9,10,11]],[[11,8,10,7,9,1,2,3,4,5,6],[2,11,8,10,7,9,3,4,5,6,1],[9,3,11,8,10,7,4,5,6,1,2],[7,9,4,11,8,10,5,6,1,2,3],[10,7,9,5,11,8,6,1,2,3,4],[8,10,7,9,6,11,1,2,3,4,5],[3,4,5,6,1,2,9,10,11,7,8],[4,5,6,1,2,3,10,11,7,8,9],[5,6,1,2,3,4,11,7,8,9,10],[6,1,2,3,4,5,7,8,9,10,11],[1,2,3,4,5,6,8,9,10,11,7]],[[11,7,9,8,10,1,4,6,5,3,2],[2,11,8,10,7,9,1,5,4,6,3],[10,3,11,7,9,8,6,2,1,5,4],[7,9,4,11,8,10,3,1,6,2,5],[9,8,10,5,11,7,2,4,3,1,6],[8,10,7,9,6,11,5,3,2,4,1],[4,1,6,3,2,5,11,9,10,7,8],[6,5,2,1,4,3,10,11,8,9,7],[5,4,1,6,3,2,9,7,11,8,10],[3,6,5,2,1,4,8,10,7,11,9],[1,2,3,4,5,6,7,8,9,10,11]],[[7,5,8,11,9,10,4,3,2,6,1],[10,8,6,9,11,7,3,5,4,1,2],[8,10,9,1,7,11,5,4,6,2,3],[11,9,10,7,2,8,1,6,5,3,4],[9,11,7,10,8,3,6,2,1,4,5],[4,7,11,8,10,9,2,1,3,5,6],[6,2,1,3,5,4,10,11,7,8,9],[5,1,3,2,4,6,8,10,11,9,7],[1,6,2,4,3,5,11,9,10,7,8],[2,3,4,5,6,1,9,7,8,10,11],[3,4,5,6,1,2,7,8,9,11,10]],[[8,11,6,10,7,9,1,2,3,4,5],[2,8,11,7,10,1,9,3,4,5,6],[9,3,8,11,1,10,2,4,5,6,7],[3,9,4,8,11,2,10,5,6,7,1],[10,4,9,5,8,11,3,6,7,1,2],[4,10,5,9,6,8,11,7,1,2,3],[11,5,10,6,9,7,8,1,2,3,4],[5,6,7,1,2,3,4,10,11,8,9],[6,7,1,2,3,4,5,11,8,9,10],[7,1,2,3,4,5,6,8,9,10,11],[1,2,3,4,5,6,7,9,10,11,8]],[[11,6,10,7,9,8,4,1,5,2,3],[2,11,7,10,8,9,1,5,6,3,4],[6,3,11,8,10,1,9,2,7,4,5],[3,7,4,11,1,10,2,9,8,5,6],[9,4,8,5,11,2,10,3,1,6,7],[4,9,5,1,6,11,3,10,2,7,8],[10,5,9,6,2,7,11,4,3,8,1],[5,10,6,9,7,3,8,11,4,1,2],[7,8,1,2,3,4,5,6,11,9,10],[8,1,2,3,4,5,6,7,9,10,11],[1,2,3,4,5,6,7,8,10,11,9]],[[11,10,2,9,3,8,4,1,5,7,6],[2,11,9,3,10,4,1,5,8,6,7],[6,3,11,10,4,9,5,2,7,1,8],[3,7,4,11,9,5,10,6,2,8,1],[7,4,8,5,11,10,6,9,1,3,2],[10,8,5,1,6,11,9,7,4,2,3],[8,9,1,6,2,7,11,10,3,5,4],[9,1,10,2,7,3,8,11,6,4,5],[1,6,3,8,5,2,7,4,11,9,10],[5,2,7,4,1,6,3,8,10,11,9],[4,5,6,7,8,1,2,3,9,10,11]],[[6,11,7,10,8,4,9,5,1,2,3],[2,7,11,8,10,9,5,1,6,3,4],[7,3,8,11,9,10,1,6,2,4,5],[3,8,4,9,11,1,10,2,7,5,6],[8,4,9,5,1,11,2,10,3,6,7],[4,9,5,1,6,2,11,3,10,7,8],[10,5,1,6,2,7,3,11,4,8,9],[5,10,6,2,7,3,8,4,11,9,1],[11,6,10,7,3,8,4,9,5,1,2],[9,1,2,3,4,5,6,7,8,10,11],[1,2,3,4,5,6,7,8,9,11,10]],[[11,7,2,8,3,9,4,10,5,1,6],[2,11,8,3,9,4,10,5,1,6,7],[7,3,11,9,4,10,5,1,6,2,8],[3,8,4,11,10,5,1,6,2,7,9],[8,4,9,5,11,1,6,2,7,3,10],[4,9,5,10,6,11,2,7,3,8,1],[9,5,10,6,1,7,11,3,8,4,2],[5,10,6,1,7,2,8,11,4,9,3],[10,6,1,7,2,8,3,9,11,5,4],[6,1,7,2,8,3,9,4,10,11,5],[1,2,3,4,5,6,7,8,9,10,11]],[[7,2,8,3,9,4,10,5,11,6,1],[2,8,3,9,4,10,5,11,6,1,7],[8,3,9,4,10,5,11,6,1,7,2],[3,9,4,10,5,11,6,1,7,2,8],[9,4,10,5,11,6,1,7,2,8,3],[4,10,5,11,6,1,7,2,8,3,9],[10,5,11,6,1,7,2,8,3,9,4],[5,11,6,1,7,2,8,3,9,4,10],[11,6,1,7,2,8,3,9,4,10,5],[6,1,7,2,8,3,9,4,10,5,11],[1,7,2,8,3,9,4,10,5,11,6]]]; AutRep[12]:=[[(),(),()],[(),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2)(3,4),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2)(3,4),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5)(6,7),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5)(6,7),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5)(6,7)(8,9),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5)(6,7)(8,9),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5)(6,7)(8,9),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5)(6,7)(8,9)(10,11),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5)(6,7)(8,9)(10,11),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6),(1,2,3)(4,5,6)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6),(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8)(9,10)(11,12)],[(1,2,3)(4,5,6),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8),(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8)(9,10)(11,12)],[(1,2,3)(4,5,6)(7,8),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8)(9,10)(11,12)],[(1,2,3)(4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8)(9,10)(11,12),(1,2,3)(4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)],[(1,2,3)(4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6)(7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7)(8,9),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7)(8,9)(10,11),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7)(8,9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7)(8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)],[(1,2,3,4,5,6),(1,2,3,4,5,6),(1,2,3,4,5,6)],[(1,2,3,4,5,6),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12)],[(1,2,3,4,5,6),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8)(9,10)],[(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6)(7,8,9)(10,11)],[(1,2,3,4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9)(10,11,12)],[(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12)],[(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)]]; LatinRep[12]:=[[[4,10,11,8,6,3,12,5,1,2,7,9],[1,9,6,7,8,4,10,3,5,11,12,2],[11,2,3,1,4,5,6,9,12,8,10,7],[2,7,5,4,10,11,3,12,6,9,8,1],[10,8,1,6,11,2,4,7,3,12,9,5],[6,5,12,3,2,7,8,1,9,10,4,11],[7,12,2,5,1,8,9,11,10,3,6,4],[12,3,8,9,5,6,7,4,11,1,2,10],[8,1,10,11,9,12,5,2,4,7,3,6],[3,11,4,12,7,9,1,10,2,6,5,8],[9,4,7,2,3,10,11,6,8,5,1,12],[5,6,9,10,12,1,2,8,7,4,11,3]],[[7,8,9,10,11,12,1,2,3,4,5,6],[11,12,8,7,10,9,4,3,6,5,1,2],[10,9,12,11,7,8,5,6,2,1,4,3],[8,7,10,9,12,11,3,4,1,2,6,5],[12,11,7,8,9,10,6,5,4,3,2,1],[9,10,11,12,8,7,2,1,5,6,3,4],[1,2,3,4,5,6,7,8,9,10,11,12],[3,4,5,6,2,1,8,7,10,9,12,11],[5,6,2,1,4,3,9,10,11,12,7,8],[2,1,4,3,6,5,10,9,12,11,8,7],[4,3,6,5,1,2,11,12,7,8,9,10],[6,5,1,2,3,4,12,11,8,7,10,9]],[[7,8,9,10,11,12,4,5,6,1,2,3],[11,12,10,9,7,8,3,1,2,5,6,4],[9,7,8,12,10,11,1,2,3,4,5,6],[10,11,12,8,9,7,5,6,4,3,1,2],[8,9,7,11,12,10,6,4,5,2,3,1],[12,10,11,7,8,9,2,3,1,6,4,5],[4,5,6,2,3,1,9,7,8,10,11,12],[1,2,3,4,5,6,7,8,9,11,12,10],[2,3,1,5,6,4,8,9,7,12,10,11],[6,4,5,1,2,3,10,11,12,7,8,9],[3,1,2,6,4,5,11,12,10,9,7,8],[5,6,4,3,1,2,12,10,11,8,9,7]],[[1,2,3,4,9,10,11,12,5,6,7,8],[12,9,10,11,5,6,7,8,2,3,4,1],[5,6,7,8,1,2,3,4,9,10,11,12],[9,10,11,12,6,7,8,5,1,2,3,4],[8,5,6,7,2,3,4,1,10,11,12,9],[2,3,4,1,12,9,10,11,6,7,8,5],[6,7,8,5,10,11,12,9,3,4,1,2],[10,11,12,9,3,4,1,2,7,8,5,6],[3,4,1,2,7,8,5,6,11,12,9,10],[7,8,5,6,11,12,9,10,4,1,2,3],[11,12,9,10,4,1,2,3,8,5,6,7],[4,1,2,3,8,5,6,7,12,9,10,11]],[[1,2,3,4,5,6,8,9,10,11,12,7],[2,3,4,5,6,1,7,8,9,10,11,12],[7,8,9,10,11,12,2,3,4,5,6,1],[8,9,10,11,12,7,3,4,5,6,1,2],[3,4,5,6,1,2,9,10,11,12,7,8],[9,10,11,12,7,8,4,5,6,1,2,3],[4,5,6,1,2,3,10,11,12,7,8,9],[10,11,12,7,8,9,5,6,1,2,3,4],[5,6,1,2,3,4,11,12,7,8,9,10],[11,12,7,8,9,10,6,1,2,3,4,5],[6,1,2,3,4,5,12,7,8,9,10,11],[12,7,8,9,10,11,1,2,3,4,5,6]],[[3,4,5,6,7,8,9,10,11,12,1,2],[4,5,6,7,8,9,10,11,12,1,2,3],[5,6,7,8,9,10,11,12,1,2,3,4],[6,7,8,9,10,11,12,1,2,3,4,5],[7,8,9,10,11,12,1,2,3,4,5,6],[8,9,10,11,12,1,2,3,4,5,6,7],[9,10,11,12,1,2,3,4,5,6,7,8],[10,11,12,1,2,3,4,5,6,7,8,9],[11,12,1,2,3,4,5,6,7,8,9,10],[12,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,12],[2,3,4,5,6,7,8,9,10,11,12,1]],[[1,6,2,4,5,3,7,12,8,10,11,9],[5,2,3,1,4,6,11,8,9,7,10,12],[3,4,5,6,2,1,9,10,11,12,8,7],[2,1,4,3,6,5,8,7,10,9,12,11],[4,3,6,5,1,2,10,9,12,11,7,8],[6,5,1,2,3,4,12,11,7,8,9,10],[7,8,9,10,11,12,1,2,3,4,5,6],[9,10,11,12,8,7,3,4,5,6,2,1],[11,12,8,7,10,9,5,6,2,1,4,3],[8,7,10,9,12,11,2,1,4,3,6,5],[10,9,12,11,7,8,4,3,6,5,1,2],[12,11,7,8,9,10,6,5,1,2,3,4]],[[1,10,3,12,2,6,4,8,9,7,11,5],[9,2,11,4,5,3,7,1,6,10,8,12],[5,6,7,8,9,10,11,12,2,3,4,1],[2,3,4,1,6,7,8,5,10,11,12,9],[6,7,8,5,10,11,12,9,3,4,1,2],[10,11,12,9,3,4,1,2,7,8,5,6],[3,4,1,2,7,8,5,6,11,12,9,10],[7,8,5,6,11,12,9,10,4,1,2,3],[11,12,9,10,4,1,2,3,8,5,6,7],[4,1,2,3,8,5,6,7,12,9,10,11],[8,5,6,7,12,9,10,11,1,2,3,4],[12,9,10,11,1,2,3,4,5,6,7,8]],[[1,4,3,6,5,2,7,10,9,12,11,8],[3,2,5,4,1,6,9,8,11,10,7,12],[2,3,4,5,6,1,8,9,10,11,12,7],[4,5,6,1,2,3,10,11,12,7,8,9],[5,6,1,2,3,4,11,12,7,8,9,10],[6,1,2,3,4,5,12,7,8,9,10,11],[7,8,9,10,11,12,1,2,3,4,5,6],[8,9,10,11,12,7,2,3,4,5,6,1],[9,10,11,12,7,8,3,4,5,6,1,2],[10,11,12,7,8,9,4,5,6,1,2,3],[11,12,7,8,9,10,5,6,1,2,3,4],[12,7,8,9,10,11,6,1,2,3,4,5]],[[1,4,3,6,5,8,7,10,9,12,11,2],[3,2,5,4,7,6,9,8,11,10,1,12],[2,3,4,5,6,7,8,9,10,11,12,1],[4,5,6,7,8,9,10,11,12,1,2,3],[5,6,7,8,9,10,11,12,1,2,3,4],[6,7,8,9,10,11,12,1,2,3,4,5],[7,8,9,10,11,12,1,2,3,4,5,6],[8,9,10,11,12,1,2,3,4,5,6,7],[9,10,11,12,1,2,3,4,5,6,7,8],[10,11,12,1,2,3,4,5,6,7,8,9],[11,12,1,2,3,4,5,6,7,8,9,10],[12,1,2,3,4,5,6,7,8,9,10,11]],[[1,11,5,12,3,6,4,10,9,8,2,7],[12,2,11,6,5,4,9,3,7,10,8,1],[9,4,3,2,7,1,5,8,6,12,11,10],[3,10,1,4,2,8,7,6,11,5,9,12],[5,6,9,10,1,2,11,12,4,3,7,8],[10,9,7,8,11,12,3,4,2,1,6,5],[7,8,12,11,9,10,2,1,5,6,4,3],[11,12,10,9,8,7,6,5,3,4,1,2],[4,3,2,1,12,11,10,9,8,7,5,6],[8,7,6,5,4,3,1,2,12,11,10,9],[2,1,8,7,6,5,12,11,10,9,3,4],[6,5,4,3,10,9,8,7,1,2,12,11]],[[1,10,3,12,2,6,4,8,9,7,11,5],[9,2,11,4,5,3,7,1,6,10,8,12],[5,9,7,11,1,10,3,12,2,6,4,8],[12,6,10,8,9,2,11,4,5,3,7,1],[2,3,4,1,6,7,8,5,10,11,12,9],[6,7,8,5,10,11,12,9,3,4,1,2],[10,11,12,9,3,4,1,2,7,8,5,6],[3,4,1,2,7,8,5,6,11,12,9,10],[7,8,5,6,11,12,9,10,4,1,2,3],[11,12,9,10,4,1,2,3,8,5,6,7],[4,1,2,3,8,5,6,7,12,9,10,11],[8,5,6,7,12,9,10,11,1,2,3,4]],[[1,7,3,9,5,11,2,10,4,12,6,8],[12,2,8,4,10,6,9,3,11,5,7,1],[3,8,5,10,1,12,7,2,9,4,11,6],[7,4,9,6,11,2,1,8,3,10,5,12],[2,3,4,5,6,1,10,11,12,7,8,9],[4,5,6,1,2,3,8,9,10,11,12,7],[8,9,10,11,12,7,4,5,6,1,2,3],[9,10,11,12,7,8,3,4,5,6,1,2],[10,11,12,7,8,9,6,1,2,3,4,5],[11,12,7,8,9,10,5,6,1,2,3,4],[5,6,1,2,3,4,12,7,8,9,10,11],[6,1,2,3,4,5,11,12,7,8,9,10]],[[1,4,3,6,5,8,7,10,9,12,11,2],[3,2,5,4,7,6,9,8,11,10,1,12],[2,1,4,3,6,5,8,7,10,9,12,11],[12,3,2,5,4,7,6,9,8,11,10,1],[4,5,6,7,8,9,10,11,12,1,2,3],[5,6,7,8,9,10,11,12,1,2,3,4],[6,7,8,9,10,11,12,1,2,3,4,5],[7,8,9,10,11,12,1,2,3,4,5,6],[8,9,10,11,12,1,2,3,4,5,6,7],[9,10,11,12,1,2,3,4,5,6,7,8],[10,11,12,1,2,3,4,5,6,7,8,9],[11,12,1,2,3,4,5,6,7,8,9,10]],[[12,8,9,10,11,7,3,5,2,4,6,1],[8,12,10,9,7,11,4,6,1,3,5,2],[7,11,12,8,9,10,5,2,4,6,1,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8,10,3,5,12,2,7,9],[10,5,7,12,2,9,11,4,6,1,3,8],[9,11,6,8,1,3,10,12,5,7,2,4],[5,10,12,7,9,2,4,11,1,6,8,3],[6,9,8,11,10,1,12,3,2,5,4,7],[8,7,10,9,12,11,2,1,4,3,6,5]],[[6,7,1,8,5,12,3,10,4,11,2,9],[7,1,8,2,9,6,10,4,11,5,12,3],[1,8,2,9,3,10,4,11,5,12,6,7],[11,2,9,3,10,4,8,5,12,6,7,1],[5,12,3,10,4,11,2,9,6,7,1,8],[12,6,7,4,11,5,9,3,10,1,8,2],[3,10,4,11,2,9,6,7,1,8,5,12],[10,4,11,5,12,3,7,1,8,2,9,6],[4,11,5,12,6,7,1,8,2,9,3,10],[8,5,12,6,7,1,11,2,9,3,10,4],[2,9,6,7,1,8,5,12,3,10,4,11],[9,3,10,1,8,2,12,6,7,4,11,5]],[[1,3,2,6,11,4,7,9,8,12,5,10],[11,2,4,3,7,12,5,8,10,9,1,6],[7,12,3,5,4,8,1,6,9,11,10,2],[3,8,1,4,6,5,9,2,7,10,12,11],[12,4,9,2,5,7,6,10,3,8,11,1],[2,1,5,10,3,6,8,7,11,4,9,12],[4,6,8,11,1,3,10,12,2,5,7,9],[10,5,7,9,12,2,4,11,1,3,6,8],[9,11,6,8,10,1,3,5,12,2,4,7],[8,10,12,7,9,11,2,4,6,1,3,5],[6,9,11,1,8,10,12,3,5,7,2,4],[5,7,10,12,2,9,11,1,4,6,8,3]],[[9,12,8,11,7,10,1,2,3,4,5,6],[2,9,12,8,11,1,10,3,4,5,6,7],[10,3,9,12,8,11,2,4,5,6,7,1],[3,10,4,9,12,8,11,5,6,7,1,2],[11,4,10,5,9,12,8,6,7,1,2,3],[8,11,5,10,6,9,12,7,1,2,3,4],[12,8,11,6,10,7,9,1,2,3,4,5],[4,5,6,7,1,2,3,10,11,12,8,9],[5,6,7,1,2,3,4,11,12,8,9,10],[6,7,1,2,3,4,5,12,8,9,10,11],[7,1,2,3,4,5,6,8,9,10,11,12],[1,2,3,4,5,6,7,9,10,11,12,8]],[[12,6,11,7,10,8,9,1,5,2,3,4],[2,12,7,11,8,10,1,9,6,3,4,5],[9,3,12,8,11,1,10,2,7,4,5,6],[3,9,4,12,1,11,2,10,8,5,6,7],[10,4,9,5,12,2,11,3,1,6,7,8],[4,10,5,9,6,12,3,11,2,7,8,1],[11,5,10,6,9,7,12,4,3,8,1,2],[5,11,6,10,7,9,8,12,4,1,2,3],[6,7,8,1,2,3,4,5,11,12,9,10],[7,8,1,2,3,4,5,6,12,9,10,11],[8,1,2,3,4,5,6,7,9,10,11,12],[1,2,3,4,5,6,7,8,10,11,12,9]],[[1,3,2,6,9,11,10,12,4,8,5,7],[11,2,4,3,7,10,12,9,1,5,8,6],[10,12,3,5,4,8,9,11,6,2,7,1],[12,9,11,4,6,5,1,10,3,7,2,8],[9,11,10,12,5,7,6,2,8,4,1,3],[3,10,12,9,11,6,8,7,5,1,4,2],[8,4,9,11,10,12,7,1,2,6,3,5],[2,1,5,10,12,9,11,8,7,3,6,4],[4,5,6,7,8,1,2,3,10,9,11,12],[7,8,1,2,3,4,5,6,9,10,12,11],[6,7,8,1,2,3,4,5,11,12,9,10],[5,6,7,8,1,2,3,4,12,11,10,9]],[[1,12,10,9,6,3,11,2,4,7,5,8],[3,2,9,11,10,7,4,12,1,5,8,6],[9,4,3,10,12,11,8,5,7,2,6,1],[6,10,5,4,11,9,12,1,2,8,3,7],[2,7,11,6,5,12,10,9,8,3,1,4],[10,3,8,12,7,6,9,11,5,1,4,2],[12,11,4,1,9,8,7,10,3,6,2,5],[11,9,12,5,2,10,1,8,6,4,7,3],[4,5,6,7,8,1,2,3,9,10,11,12],[5,6,7,8,1,2,3,4,10,11,12,9],[7,8,1,2,3,4,5,6,11,12,9,10],[8,1,2,3,4,5,6,7,12,9,10,11]],[[12,10,11,9,3,8,4,1,5,7,6,2],[2,12,9,11,10,4,1,5,8,6,7,3],[6,3,12,10,11,9,5,2,7,1,8,4],[3,7,4,12,9,11,10,6,2,8,1,5],[7,4,8,5,12,10,11,9,1,3,2,6],[10,8,5,1,6,12,9,11,4,2,3,7],[11,9,1,6,2,7,12,10,3,5,4,8],[9,11,10,2,7,3,8,12,6,4,5,1],[1,6,3,8,5,2,7,4,12,11,10,9],[5,2,7,4,1,6,3,8,11,12,9,10],[4,5,6,7,8,1,2,3,10,9,11,12],[8,1,2,3,4,5,6,7,9,10,12,11]],[[1,3,2,6,9,11,10,12,4,8,5,7],[11,2,4,3,7,10,12,9,1,5,8,6],[10,12,3,5,4,8,9,11,6,2,7,1],[12,9,11,4,6,5,1,10,3,7,2,8],[9,11,10,12,5,7,6,2,8,4,1,3],[3,10,12,9,11,6,8,7,5,1,4,2],[8,4,9,11,10,12,7,1,2,6,3,5],[2,1,5,10,12,9,11,8,7,3,6,4],[4,7,6,1,8,3,2,5,11,9,10,12],[6,5,8,7,2,1,4,3,10,12,11,9],[5,6,7,8,1,2,3,4,9,10,12,11],[7,8,1,2,3,4,5,6,12,11,9,10]],[[1,3,6,9,11,2,10,12,4,7,5,8],[9,2,4,7,10,12,3,11,1,5,8,6],[12,10,3,5,8,11,9,4,7,2,6,1],[5,9,11,4,6,1,12,10,2,8,3,7],[11,6,10,12,5,7,2,9,8,3,1,4],[10,12,7,11,9,6,8,3,5,1,4,2],[4,11,9,8,12,10,7,1,3,6,2,5],[2,5,12,10,1,9,11,8,6,4,7,3],[3,8,5,2,7,4,1,6,10,9,12,11],[7,4,1,6,3,8,5,2,12,11,10,9],[8,1,2,3,4,5,6,7,9,10,11,12],[6,7,8,1,2,3,4,5,11,12,9,10]],[[12,10,3,9,4,8,11,1,7,5,2,6],[2,11,9,4,10,5,1,12,6,8,7,3],[11,3,12,10,5,9,6,2,1,7,4,8],[3,12,4,11,9,6,10,7,8,2,1,5],[8,4,11,5,12,10,7,9,3,1,6,2],[10,1,5,12,6,11,9,8,2,4,3,7],[1,9,2,6,11,7,12,10,5,3,8,4],[9,2,10,3,7,12,8,11,4,6,5,1],[7,6,1,8,3,2,5,4,9,12,10,11],[5,8,7,2,1,4,3,6,11,10,12,9],[4,7,6,1,8,3,2,5,10,11,9,12],[6,5,8,7,2,1,4,3,12,9,11,10]],[[1,12,10,9,6,3,11,2,4,7,5,8],[3,2,9,11,10,7,4,12,1,5,8,6],[9,4,3,10,12,11,8,5,7,2,6,1],[6,10,5,4,11,9,12,1,2,8,3,7],[2,7,11,6,5,12,10,9,8,3,1,4],[10,3,8,12,7,6,9,11,5,1,4,2],[12,11,4,1,9,8,7,10,3,6,2,5],[11,9,12,5,2,10,1,8,6,4,7,3],[4,1,6,3,8,5,2,7,9,12,11,10],[8,5,2,7,4,1,6,3,11,10,9,12],[5,8,7,2,1,4,3,6,10,9,12,11],[7,6,1,8,3,2,5,4,12,11,10,9]],[[6,12,7,11,8,10,9,5,1,4,3,2],[2,7,12,8,11,9,10,1,6,5,4,3],[7,3,8,12,9,11,1,10,2,6,5,4],[3,8,4,9,12,1,11,2,10,7,6,5],[10,4,9,5,1,12,2,11,3,8,7,6],[4,10,5,1,6,2,12,3,11,9,8,7],[11,5,10,6,2,7,3,12,4,1,9,8],[5,11,6,10,7,3,8,4,12,2,1,9],[12,6,11,7,10,8,4,9,5,3,2,1],[1,2,3,4,5,6,7,8,9,12,10,11],[9,1,2,3,4,5,6,7,8,11,12,10],[8,9,1,2,3,4,5,6,7,10,11,12]],[[1,6,2,10,3,11,4,12,5,8,7,9],[6,2,7,3,11,4,12,5,10,1,9,8],[11,7,3,8,4,12,5,10,6,9,2,1],[7,12,8,4,9,5,10,6,11,2,1,3],[12,8,10,9,5,1,6,11,7,4,3,2],[8,10,9,11,1,6,2,7,12,3,5,4],[10,9,11,1,12,2,7,3,8,5,4,6],[9,11,1,12,2,10,3,8,4,7,6,5],[5,1,12,2,10,3,11,4,9,6,8,7],[2,3,4,5,6,7,8,9,1,10,11,12],[3,4,5,6,7,8,9,1,2,11,12,10],[4,5,6,7,8,9,1,2,3,12,10,11]],[[6,11,7,12,8,10,9,5,1,3,2,4],[2,7,12,8,10,9,11,1,6,5,4,3],[7,3,8,10,9,11,1,12,2,4,6,5],[3,8,4,9,11,1,12,2,10,6,5,7],[11,4,9,5,1,12,2,10,3,8,7,6],[4,12,5,1,6,2,10,3,11,7,9,8],[12,5,10,6,2,7,3,11,4,9,8,1],[5,10,6,11,7,3,8,4,12,2,1,9],[10,6,11,7,12,8,4,9,5,1,3,2],[9,2,1,3,5,4,6,8,7,12,11,10],[8,1,3,2,4,6,5,7,9,11,10,12],[1,9,2,4,3,5,7,6,8,10,12,11]],[[12,11,3,8,4,9,5,10,6,1,7,2],[2,12,11,4,9,5,10,6,1,7,8,3],[8,3,12,11,5,10,6,1,7,2,9,4],[3,9,4,12,11,6,1,7,2,8,10,5],[9,4,10,5,12,11,7,2,8,3,1,6],[4,10,5,1,6,12,11,8,3,9,2,7],[10,5,1,6,2,7,12,11,9,4,3,8],[5,1,6,2,7,3,8,12,11,10,4,9],[1,6,2,7,3,8,4,9,12,11,5,10],[11,2,7,3,8,4,9,5,10,12,6,1],[7,8,9,10,1,2,3,4,5,6,12,11],[6,7,8,9,10,1,2,3,4,5,11,12]],[[1,4,2,7,3,10,8,11,6,12,5,9],[11,2,5,3,8,4,1,9,12,7,10,6],[8,12,3,6,4,9,5,2,10,11,7,1],[12,9,11,4,7,5,10,6,3,1,2,8],[2,11,10,12,5,8,6,1,7,4,9,3],[5,3,12,1,11,6,9,7,2,8,4,10],[9,6,4,11,2,12,7,10,8,3,1,5],[4,10,7,5,12,3,11,8,1,9,6,2],[10,5,1,8,6,11,4,12,9,2,3,7],[3,1,6,2,9,7,12,5,11,10,8,4],[6,7,8,9,10,1,2,3,4,5,11,12],[7,8,9,10,1,2,3,4,5,6,12,11]],[[7,12,8,3,9,4,10,5,11,6,1,2],[2,8,12,9,4,10,5,11,6,1,7,3],[8,3,9,12,10,5,11,6,1,7,2,4],[3,9,4,10,12,11,6,1,7,2,8,5],[9,4,10,5,11,12,1,7,2,8,3,6],[4,10,5,11,6,1,12,2,8,3,9,7],[10,5,11,6,1,7,2,12,3,9,4,8],[5,11,6,1,7,2,8,3,12,4,10,9],[11,6,1,7,2,8,3,9,4,12,5,10],[6,1,7,2,8,3,9,4,10,5,12,11],[12,7,2,8,3,9,4,10,5,11,6,1],[1,2,3,4,5,6,7,8,9,10,11,12]]]; AutRep[13]:=[[(),(),()],[(),(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6)(7,8,9)(10,11),(1,2,3,4,5,6)(7,8,9)(10,11)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13)]]; LatinRep[13]:=[[[6,7,10,2,12,4,1,9,8,11,13,5,3],[4,1,7,8,10,9,11,3,13,5,12,2,6],[3,11,2,4,7,1,9,8,6,12,5,13,10],[7,10,3,13,5,12,8,1,11,4,9,6,2],[8,5,4,10,9,6,13,7,3,1,2,12,11],[9,3,13,6,4,2,7,11,5,8,10,1,12],[5,2,8,3,13,10,6,4,12,9,11,7,1],[13,12,11,5,1,3,4,2,10,6,7,9,8],[12,13,1,11,8,7,3,6,9,2,4,10,5],[11,9,6,7,2,13,5,12,1,10,3,8,4],[1,8,12,9,11,5,10,13,2,3,6,4,7],[10,6,9,12,3,8,2,5,4,7,1,11,13],[2,4,5,1,6,11,12,10,7,13,8,3,9]],[[3,4,5,6,7,8,9,10,11,12,13,1,2],[4,5,6,7,8,9,10,11,12,13,1,2,3],[5,6,7,8,9,10,11,12,13,1,2,3,4],[6,7,8,9,10,11,12,13,1,2,3,4,5],[7,8,9,10,11,12,13,1,2,3,4,5,6],[8,9,10,11,12,13,1,2,3,4,5,6,7],[9,10,11,12,13,1,2,3,4,5,6,7,8],[10,11,12,13,1,2,3,4,5,6,7,8,9],[11,12,13,1,2,3,4,5,6,7,8,9,10],[12,13,1,2,3,4,5,6,7,8,9,10,11],[13,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,13],[2,3,4,5,6,7,8,9,10,11,12,13,1]],[[13,5,10,12,3,7,11,9,2,8,4,1,6],[6,13,12,10,8,4,9,11,1,7,3,2,5],[10,12,13,7,11,9,5,2,3,4,1,6,8],[12,10,8,13,9,11,1,6,4,3,2,5,7],[4,7,11,9,13,2,12,10,8,5,6,3,1],[8,3,9,11,1,13,10,12,7,6,5,4,2],[11,9,6,2,12,10,13,4,5,1,7,8,3],[9,11,1,5,10,12,3,13,6,2,8,7,4],[1,2,4,3,7,8,6,5,13,12,10,9,11],[7,8,3,4,6,5,2,1,10,13,9,11,12],[3,4,2,1,5,6,8,7,12,11,13,10,9],[2,1,5,6,4,3,7,8,11,9,12,13,10],[5,6,7,8,2,1,4,3,9,10,11,12,13]],[[13,8,11,9,5,10,6,12,7,1,4,2,3],[7,13,10,11,9,6,12,5,2,8,3,1,4],[3,9,13,10,11,2,7,1,8,12,6,4,5],[10,4,9,13,1,11,2,8,12,7,5,3,6],[12,10,5,2,13,1,11,4,9,3,8,6,7],[9,12,1,6,2,13,3,11,4,10,7,5,8],[5,2,12,1,7,4,13,3,11,6,10,8,9],[1,6,2,12,3,8,4,13,5,11,9,7,10],[8,11,7,4,12,3,9,6,13,5,1,10,2],[11,7,3,8,4,12,5,10,6,13,2,9,1],[4,3,6,5,8,7,10,9,1,2,13,11,12],[2,1,4,3,6,5,8,7,10,9,12,13,11],[6,5,8,7,10,9,1,2,3,4,11,12,13]],[[13,7,4,10,3,9,6,12,5,11,8,1,2],[8,13,9,3,10,4,11,5,12,6,2,7,1],[3,10,13,9,6,12,5,11,8,1,7,2,4],[9,4,10,13,11,5,12,6,2,7,1,8,3],[4,9,5,12,13,11,8,1,7,2,10,3,6],[10,3,11,6,12,13,2,7,1,8,4,9,5],[5,12,6,11,7,1,13,2,10,3,9,4,8],[11,6,12,5,2,8,1,13,4,9,3,10,7],[6,11,7,1,8,2,9,3,13,4,12,5,10],[12,5,2,8,1,7,4,10,3,13,6,11,9],[7,1,8,2,9,3,10,4,11,5,13,6,12],[2,8,1,7,4,10,3,9,6,12,5,13,11],[1,2,3,4,5,6,7,8,9,10,11,12,13]],[[8,12,9,11,6,5,3,10,13,7,4,2,1],[7,9,12,6,11,4,13,1,10,8,5,3,2],[12,8,7,5,4,11,10,13,2,9,6,1,3],[13,6,11,3,10,1,12,9,8,5,2,7,4],[11,13,4,2,1,10,9,12,7,6,3,8,5],[5,11,13,10,3,2,8,7,12,4,1,9,6],[3,10,1,13,9,12,6,11,4,2,8,5,7],[2,1,10,12,13,7,5,4,11,3,9,6,8],[10,3,2,8,12,13,11,6,5,1,7,4,9],[9,7,8,4,5,6,1,2,3,12,13,10,11],[6,4,5,1,2,3,7,8,9,11,10,13,12],[1,2,3,9,7,8,4,5,6,13,12,11,10],[4,5,6,7,8,9,2,3,1,10,11,12,13]],[[13,7,2,11,6,12,4,10,5,3,9,1,8],[3,13,8,10,12,4,6,5,11,2,1,7,9],[9,1,13,5,11,10,12,4,6,8,3,2,7],[4,6,12,13,10,5,3,9,1,7,2,8,11],[10,5,4,6,13,11,2,1,7,9,8,3,12],[5,11,6,12,4,13,8,3,2,1,7,9,10],[11,10,5,7,9,1,13,2,8,6,12,4,3],[6,12,11,2,8,7,9,13,3,5,4,10,1],[12,4,10,8,3,9,1,7,13,11,6,5,2],[7,9,1,3,2,8,10,12,4,13,5,11,6],[2,8,7,9,1,3,5,11,10,12,13,6,4],[8,3,9,1,7,2,11,6,12,4,10,13,5],[1,2,3,4,5,6,7,8,9,10,11,12,13]],[[13,2,12,11,10,9,8,1,4,3,6,7,5],[11,13,3,12,2,10,9,5,1,4,7,8,6],[12,11,13,4,6,3,10,9,2,1,8,5,7],[1,12,11,13,9,7,4,10,3,2,5,6,8],[5,9,10,1,13,6,11,12,7,8,4,3,2],[2,6,9,10,12,13,7,11,8,5,1,4,3],[10,3,7,9,11,12,13,8,5,6,2,1,4],[9,10,4,8,5,11,12,13,6,7,3,2,1],[4,1,2,3,7,8,5,6,13,11,12,9,10],[3,4,1,2,8,5,6,7,12,13,10,11,9],[6,7,8,5,4,1,2,3,11,9,13,10,12],[7,8,5,6,3,4,1,2,10,12,9,13,11],[8,5,6,7,1,2,3,4,9,10,11,12,13]],[[13,5,2,6,11,10,12,9,3,7,8,1,4],[7,13,6,3,10,12,9,11,8,4,2,5,1],[4,8,13,7,12,9,11,10,1,5,6,3,2],[8,1,5,13,9,11,10,12,6,2,4,7,3],[11,10,12,9,13,2,6,3,5,1,7,4,8],[10,12,9,11,4,13,3,7,2,6,1,8,5],[12,9,11,10,8,1,13,4,7,3,5,2,6],[9,11,10,12,1,5,2,13,4,8,3,6,7],[1,6,3,8,7,4,5,2,13,11,12,9,10],[5,2,7,4,3,8,1,6,12,13,10,11,9],[6,4,8,2,5,3,7,1,11,9,13,10,12],[3,7,1,5,2,6,4,8,10,12,9,13,11],[2,3,4,1,6,7,8,5,9,10,11,12,13]],[[13,11,2,12,7,9,8,10,5,4,6,1,3],[9,13,12,3,11,8,10,5,2,6,1,7,4],[4,10,13,9,6,12,5,11,8,3,7,2,1],[10,1,11,13,12,7,9,6,3,5,4,8,2],[5,9,8,10,13,4,6,1,11,2,12,3,7],[11,6,10,5,2,13,1,7,4,12,3,9,8],[6,12,7,11,8,3,13,2,10,1,9,4,5],[12,7,9,8,3,5,4,13,1,11,2,10,6],[7,4,6,1,9,2,12,3,13,8,10,5,11],[2,8,1,7,4,10,3,9,6,13,5,11,12],[8,3,5,2,10,1,11,4,12,7,13,6,9],[3,5,4,6,1,11,2,12,7,9,8,13,10],[1,2,3,4,5,6,7,8,9,10,11,12,13]],[[13,6,2,7,3,11,12,5,10,1,9,8,4],[4,13,7,3,8,2,11,12,1,6,10,9,5],[9,5,13,8,4,7,3,11,12,2,6,10,1],[5,10,1,13,9,3,8,4,11,12,7,6,2],[10,1,6,2,13,12,4,9,5,11,8,7,3],[6,12,11,10,1,13,2,7,3,8,4,5,9],[2,7,12,11,6,9,13,3,8,4,5,1,10],[7,3,8,12,11,5,10,13,4,9,1,2,6],[11,8,4,9,12,10,1,6,13,5,2,3,7],[12,11,9,5,10,1,6,2,7,13,3,4,8],[1,2,3,4,5,8,9,10,6,7,13,11,12],[3,4,5,1,2,6,7,8,9,10,12,13,11],[8,9,10,6,7,4,5,1,2,3,11,12,13]],[[12,8,13,9,11,1,4,6,10,7,5,2,3],[2,12,9,13,7,10,11,5,1,6,8,3,4],[11,3,12,7,13,8,2,10,6,9,1,4,5],[9,10,4,12,8,13,1,3,11,2,7,5,6],[13,7,11,5,12,9,10,2,4,8,3,6,1],[7,13,8,10,6,12,5,11,3,4,9,1,2],[10,4,6,11,1,3,12,13,7,5,2,8,9],[4,11,5,1,10,2,8,12,13,3,6,9,7],[3,5,10,6,2,11,13,9,12,1,4,7,8],[5,9,1,8,3,7,6,4,2,13,12,11,10],[8,6,7,2,9,4,3,1,5,12,13,10,11],[6,1,2,3,4,5,9,7,8,11,10,12,13],[1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11,5,12,13,6],[1,2,3,4,5,6,7,8,9,10,11,12,13]],[[8,2,9,3,10,4,11,5,12,6,13,7,1],[2,9,3,10,4,11,5,12,6,13,7,1,8],[9,3,10,4,11,5,12,6,13,7,1,8,2],[3,10,4,11,5,12,6,13,7,1,8,2,9],[10,4,11,5,12,6,13,7,1,8,2,9,3],[4,11,5,12,6,13,7,1,8,2,9,3,10],[11,5,12,6,13,7,1,8,2,9,3,10,4],[5,12,6,13,7,1,8,2,9,3,10,4,11],[12,6,13,7,1,8,2,9,3,10,4,11,5],[6,13,7,1,8,2,9,3,10,4,11,5,12],[13,7,1,8,2,9,3,10,4,11,5,12,6],[7,1,8,2,9,3,10,4,11,5,12,6,13],[1,8,2,9,3,10,4,11,5,12,6,13,7]]]; AutRep[14]:=[[(),(),()],[(),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)],[(),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2)(3,4),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5)(6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5)(6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)],[(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7)(8,9),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7)(8,9)(10,11),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7)(8,9)(10,11)(12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)],[(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13)]]; LatinRep[14]:=[[[4,11,2,7,13,9,10,12,8,5,14,3,1,6],[6,7,12,13,3,5,4,14,11,2,9,1,10,8],[2,1,6,9,14,7,8,5,10,3,13,12,11,4],[3,9,8,1,4,6,11,13,7,12,10,2,5,14],[14,8,1,12,10,4,7,2,9,11,3,5,6,13],[10,13,5,2,7,12,14,4,1,8,6,11,9,3],[11,2,4,8,9,1,5,6,3,7,12,14,13,10],[8,4,14,11,1,2,12,3,13,6,5,10,7,9],[13,14,9,3,12,8,2,1,5,10,11,6,4,7],[9,10,3,5,11,14,6,7,2,13,4,8,12,1],[5,3,13,14,6,10,9,8,12,4,1,7,2,11],[7,12,10,6,2,3,1,11,4,9,8,13,14,5],[12,6,11,4,5,13,3,10,14,1,7,9,8,2],[1,5,7,10,8,11,13,9,6,14,2,4,3,12]],[[1,2,5,6,3,4,11,12,13,14,7,8,9,10],[13,14,3,4,7,8,5,6,12,11,10,9,1,2],[12,11,14,13,5,6,9,10,7,8,2,1,4,3],[10,9,11,12,13,14,7,8,2,1,3,4,5,6],[3,4,1,2,12,11,14,13,9,10,6,5,8,7],[5,6,9,10,1,2,12,11,14,13,4,3,7,8],[14,13,7,8,2,1,3,4,11,12,5,6,10,9],[11,12,13,14,9,10,4,3,5,6,8,7,2,1],[8,7,12,11,14,13,2,1,6,5,9,10,3,4],[7,8,10,9,11,12,13,14,4,3,1,2,6,5],[9,10,2,1,4,3,6,5,8,7,12,11,13,14],[6,5,8,7,10,9,1,2,3,4,11,12,14,13],[4,3,6,5,8,7,10,9,1,2,13,14,11,12],[2,1,4,3,6,5,8,7,10,9,14,13,12,11]],[[1,2,3,4,5,6,7,9,10,11,12,13,14,8],[2,3,4,5,6,7,1,8,9,10,11,12,13,14],[8,9,10,11,12,13,14,2,3,4,5,6,7,1],[9,10,11,12,13,14,8,3,4,5,6,7,1,2],[3,4,5,6,7,1,2,10,11,12,13,14,8,9],[10,11,12,13,14,8,9,4,5,6,7,1,2,3],[4,5,6,7,1,2,3,11,12,13,14,8,9,10],[11,12,13,14,8,9,10,5,6,7,1,2,3,4],[5,6,7,1,2,3,4,12,13,14,8,9,10,11],[12,13,14,8,9,10,11,6,7,1,2,3,4,5],[6,7,1,2,3,4,5,13,14,8,9,10,11,12],[13,14,8,9,10,11,12,7,1,2,3,4,5,6],[7,1,2,3,4,5,6,14,8,9,10,11,12,13],[14,8,9,10,11,12,13,1,2,3,4,5,6,7]],[[3,4,5,6,7,8,9,10,11,12,13,14,1,2],[4,5,6,7,8,9,10,11,12,13,14,1,2,3],[5,6,7,8,9,10,11,12,13,14,1,2,3,4],[6,7,8,9,10,11,12,13,14,1,2,3,4,5],[7,8,9,10,11,12,13,14,1,2,3,4,5,6],[8,9,10,11,12,13,14,1,2,3,4,5,6,7],[9,10,11,12,13,14,1,2,3,4,5,6,7,8],[10,11,12,13,14,1,2,3,4,5,6,7,8,9],[11,12,13,14,1,2,3,4,5,6,7,8,9,10],[12,13,14,1,2,3,4,5,6,7,8,9,10,11],[13,14,1,2,3,4,5,6,7,8,9,10,11,12],[14,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[2,3,4,5,6,7,8,9,10,11,12,13,14,1]],[[1,6,7,4,5,10,11,8,9,14,2,12,13,3],[5,2,3,8,9,6,7,12,13,10,11,1,4,14],[3,4,5,6,7,8,9,10,11,12,13,14,2,1],[7,8,9,10,11,12,13,14,2,1,4,3,6,5],[9,10,11,12,13,14,2,1,4,3,6,5,8,7],[11,12,13,14,2,1,4,3,6,5,8,7,10,9],[13,14,2,1,4,3,6,5,8,7,10,9,12,11],[2,1,4,3,6,5,8,7,10,9,12,11,14,13],[4,3,6,5,8,7,10,9,12,11,14,13,1,2],[6,5,8,7,10,9,12,11,14,13,1,2,3,4],[8,7,10,9,12,11,14,13,1,2,3,4,5,6],[10,9,12,11,14,13,1,2,3,4,5,6,7,8],[12,11,14,13,1,2,3,4,5,6,7,8,9,10],[14,13,1,2,3,4,5,6,7,8,9,10,11,12]],[[1,4,3,6,5,8,7,10,9,12,11,14,13,2],[3,2,5,4,7,6,9,8,11,10,13,12,1,14],[2,3,4,5,6,7,8,9,10,11,12,13,14,1],[4,5,6,7,8,9,10,11,12,13,14,1,2,3],[5,6,7,8,9,10,11,12,13,14,1,2,3,4],[6,7,8,9,10,11,12,13,14,1,2,3,4,5],[7,8,9,10,11,12,13,14,1,2,3,4,5,6],[8,9,10,11,12,13,14,1,2,3,4,5,6,7],[9,10,11,12,13,14,1,2,3,4,5,6,7,8],[10,11,12,13,14,1,2,3,4,5,6,7,8,9],[11,12,13,14,1,2,3,4,5,6,7,8,9,10],[12,13,14,1,2,3,4,5,6,7,8,9,10,11],[13,14,1,2,3,4,5,6,7,8,9,10,11,12],[14,1,2,3,4,5,6,7,8,9,10,11,12,13]],[[1,6,7,4,5,10,11,8,9,14,2,12,13,3],[5,2,3,8,9,6,7,12,13,10,11,1,4,14],[3,13,1,6,7,4,5,10,11,8,9,14,2,12],[14,4,5,2,3,8,9,6,7,12,13,10,11,1],[7,8,9,10,11,12,13,14,2,1,4,3,6,5],[9,10,11,12,13,14,2,1,4,3,6,5,8,7],[11,12,13,14,2,1,4,3,6,5,8,7,10,9],[13,14,2,1,4,3,6,5,8,7,10,9,12,11],[2,1,4,3,6,5,8,7,10,9,12,11,14,13],[4,3,6,5,8,7,10,9,12,11,14,13,1,2],[6,5,8,7,10,9,12,11,14,13,1,2,3,4],[8,7,10,9,12,11,14,13,1,2,3,4,5,6],[10,9,12,11,14,13,1,2,3,4,5,6,7,8],[12,11,14,13,1,2,3,4,5,6,7,8,9,10]],[[1,4,3,6,5,8,7,10,9,12,11,14,13,2],[3,2,5,4,7,6,9,8,11,10,13,12,1,14],[2,1,4,3,6,5,8,7,10,9,12,11,14,13],[14,3,2,5,4,7,6,9,8,11,10,13,12,1],[4,5,6,7,8,9,10,11,12,13,14,1,2,3],[5,6,7,8,9,10,11,12,13,14,1,2,3,4],[6,7,8,9,10,11,12,13,14,1,2,3,4,5],[7,8,9,10,11,12,13,14,1,2,3,4,5,6],[8,9,10,11,12,13,14,1,2,3,4,5,6,7],[9,10,11,12,13,14,1,2,3,4,5,6,7,8],[10,11,12,13,14,1,2,3,4,5,6,7,8,9],[11,12,13,14,1,2,3,4,5,6,7,8,9,10],[12,13,14,1,2,3,4,5,6,7,8,9,10,11],[13,14,1,2,3,4,5,6,7,8,9,10,11,12]],[[1,6,7,4,5,10,11,8,9,14,2,12,13,3],[5,2,3,8,9,6,7,12,13,10,11,1,4,14],[3,13,1,6,7,4,5,10,11,8,9,14,2,12],[14,4,5,2,3,8,9,6,7,12,13,10,11,1],[7,12,13,10,11,1,4,14,2,5,8,3,6,9],[11,8,9,14,2,12,13,3,6,1,4,7,10,5],[9,10,11,12,13,14,2,1,4,3,6,5,8,7],[13,14,2,1,4,3,6,5,8,7,10,9,12,11],[2,1,4,3,6,5,8,7,10,9,12,11,14,13],[4,3,6,5,8,7,10,9,12,11,14,13,1,2],[6,5,8,7,10,9,12,11,14,13,1,2,3,4],[8,7,10,9,12,11,14,13,1,2,3,4,5,6],[10,9,12,11,14,13,1,2,3,4,5,6,7,8],[12,11,14,13,1,2,3,4,5,6,7,8,9,10]],[[1,4,3,6,5,8,7,10,9,12,11,14,13,2],[3,2,5,4,7,6,9,8,11,10,13,12,1,14],[2,1,4,3,6,5,8,7,10,9,12,11,14,13],[14,3,2,5,4,7,6,9,8,11,10,13,12,1],[4,7,6,9,8,11,10,13,12,1,14,3,2,5],[6,5,8,7,10,9,12,11,14,13,2,1,4,3],[5,6,7,8,9,10,11,12,13,14,1,2,3,4],[7,8,9,10,11,12,13,14,1,2,3,4,5,6],[8,9,10,11,12,13,14,1,2,3,4,5,6,7],[9,10,11,12,13,14,1,2,3,4,5,6,7,8],[10,11,12,13,14,1,2,3,4,5,6,7,8,9],[11,12,13,14,1,2,3,4,5,6,7,8,9,10],[12,13,14,1,2,3,4,5,6,7,8,9,10,11],[13,14,1,2,3,4,5,6,7,8,9,10,11,12]],[[14,12,13,11,10,8,9,1,5,3,7,2,6,4],[12,14,11,13,7,10,2,9,6,4,8,1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,[4,1,6,3,8,5,2,7,9,12,13,10,11,14],[7,8,1,2,3,4,5,6,14,13,12,11,10,9],[1,2,3,4,5,6,7,8,13,14,11,12,9,10]],[[14,10,13,11,3,12,4,9,5,8,6,1,7,2],[10,14,11,13,12,4,9,5,2,6,1,7,8,3],[6,11,14,12,13,9,5,10,8,3,7,2,1,4],[11,7,12,14,9,13,10,6,3,1,4,8,2,5],[7,12,8,9,14,10,13,11,1,4,2,5,3,6],[12,8,9,1,10,14,11,13,6,2,5,3,4,7],[13,9,1,10,2,11,14,12,4,7,3,6,5,8],[9,13,10,2,11,3,12,14,7,5,8,4,6,1],[1,6,4,7,5,2,8,3,14,12,13,9,11,10],[4,2,7,5,8,6,3,1,10,14,9,13,12,11],[2,5,3,8,6,1,7,4,13,11,14,10,9,12],[5,3,6,4,1,7,2,8,11,13,12,14,10,9],[3,4,5,6,7,8,1,2,12,9,10,11,13,14],[8,1,2,3,4,5,6,7,9,10,11,12,14,13]],[[10,14,7,13,8,12,9,11,1,2,3,4,5,6],[2,10,14,8,13,9,12,1,11,3,4,5,6,7],[11,3,10,14,9,13,1,12,2,4,5,6,7,8],[3,11,4,10,14,1,13,2,12,5,6,7,8,9],[12,4,11,5,10,14,2,13,3,6,7,8,9,1],[4,12,5,11,6,10,14,3,13,7,8,9,1,2],[13,5,12,6,11,7,10,14,4,8,9,1,2,3],[5,13,6,12,7,11,8,10,14,9,1,2,3,4],[14,6,13,7,12,8,11,9,10,1,2,3,4,5],[6,7,8,9,1,2,3,4,5,12,13,14,10,11],[7,8,9,1,2,3,4,5,6,13,14,10,11,12],[8,9,1,2,3,4,5,6,7,14,10,11,12,13],[9,1,2,3,4,5,6,7,8,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,11,12,13,14,10]],[[6,11,7,12,8,10,14,5,13,3,2,4,9,1],[13,7,12,8,10,9,11,14,6,5,4,3,1,2],[7,13,8,10,9,11,1,12,14,4,6,5,2,3],[14,8,13,9,11,1,12,2,10,6,5,7,3,4],[11,14,9,13,1,12,2,10,3,8,7,6,4,5],[4,12,14,1,13,2,10,3,11,7,9,8,5,6],[12,5,10,14,2,13,3,11,4,9,8,1,6,7],[5,10,6,11,14,3,13,4,12,2,1,9,7,8],[10,6,11,7,12,14,4,13,5,1,3,2,8,9],[9,2,1,3,5,4,6,8,7,13,14,10,11,12],[8,1,3,2,4,6,5,7,9,11,13,14,12,10],[1,9,2,4,3,5,7,6,8,14,12,13,10,11],[2,3,4,5,6,7,8,9,1,12,10,11,13,14],[3,4,5,6,7,8,9,1,2,10,11,12,14,13]],[[14,7,13,8,12,9,11,10,5,1,6,2,3,4],[2,14,8,13,9,12,10,11,1,6,7,3,4,5],[7,3,14,9,13,10,12,1,11,2,8,4,5,6],[3,8,4,14,10,13,1,12,2,11,9,5,6,7],[11,4,9,5,14,1,13,2,12,3,10,6,7,8],[4,11,5,10,6,14,2,13,3,12,1,7,8,9],[12,5,11,6,1,7,14,3,13,4,2,8,9,10],[5,12,6,11,7,2,8,14,4,13,3,9,10,1],[13,6,12,7,11,8,3,9,14,5,4,10,1,2],[6,13,7,12,8,11,9,4,10,14,5,1,2,3],[8,9,10,1,2,3,4,5,6,7,13,14,11,12],[9,10,1,2,3,4,5,6,7,8,14,11,12,13],[10,1,2,3,4,5,6,7,8,9,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,12,13,14,11]],[[1,3,2,6,8,11,13,5,14,12,4,10,7,9],[11,2,4,3,7,9,12,14,6,13,1,5,10,8],[14,12,3,5,4,8,10,11,13,7,6,2,9,1],[8,13,11,4,6,5,9,1,12,14,3,7,2,10],[13,9,14,12,5,7,6,10,2,11,8,4,1,3],[12,14,10,13,11,6,8,7,1,3,5,9,4,2],[4,11,13,1,14,12,7,9,8,2,10,6,3,5],[3,5,12,14,2,13,11,8,10,9,7,1,6,4],[10,4,6,11,13,3,14,12,9,1,2,8,5,7],[2,1,5,7,12,14,4,13,11,10,9,3,8,6],[5,6,7,8,9,10,1,2,3,4,12,11,13,14],[9,10,1,2,3,4,5,6,7,8,11,12,14,13],[7,8,9,10,1,2,3,4,5,6,13,14,11,12],[6,7,8,9,10,1,2,3,4,5,14,13,12,11]],[[14,11,13,12,3,9,4,10,5,1,8,6,7,2],[2,14,12,13,11,4,10,5,1,6,7,9,8,3],[7,3,14,11,13,12,5,1,6,2,10,8,9,4],[3,8,4,14,12,13,11,6,2,7,9,1,10,5],[8,4,9,5,14,11,13,12,7,3,2,10,1,6],[4,9,5,10,6,14,12,13,11,8,1,3,2,7],[9,5,10,6,1,7,14,11,13,12,4,2,3,8],[11,10,6,1,7,2,8,14,12,13,3,5,4,9],[13,12,1,7,2,8,3,9,14,11,6,4,5,10],[12,13,11,2,8,3,9,4,10,14,5,7,6,1],[1,6,3,8,5,10,7,2,9,4,14,13,12,11],[5,2,7,4,9,6,1,8,3,10,13,14,11,12],[6,7,8,9,10,1,2,3,4,5,12,11,13,14],[10,1,2,3,4,5,6,7,8,9,11,12,14,13]],[[1,3,2,6,8,11,13,5,14,12,4,10,7,9],[11,2,4,3,7,9,12,14,6,13,1,5,10,8],[14,12,3,5,4,8,10,11,13,7,6,2,9,1],[8,13,11,4,6,5,9,1,12,14,3,7,2,10],[13,9,14,12,5,7,6,10,2,11,8,4,1,3],[12,14,10,13,11,6,8,7,1,3,5,9,4,2],[4,11,13,1,14,12,7,9,8,2,10,6,3,5],[3,5,12,14,2,13,11,8,10,9,7,1,6,4],[10,4,6,11,13,3,14,12,9,1,2,8,5,7],[2,1,5,7,12,14,4,13,11,10,9,3,8,6],[5,8,7,10,9,2,1,4,3,6,13,11,12,14],[7,6,9,8,1,10,3,2,5,4,12,14,13,11],[6,7,8,9,10,1,2,3,4,5,11,12,14,13],[9,10,1,2,3,4,5,6,7,8,14,13,11,12]],[[7,14,8,13,9,12,10,5,11,6,1,4,3,2],[2,8,14,9,13,10,12,11,6,1,7,5,4,3],[8,3,9,14,10,13,11,12,1,7,2,6,5,4],[3,9,4,10,14,11,13,1,12,2,8,7,6,5],[9,4,10,5,11,14,1,13,2,12,3,8,7,6],[4,10,5,11,6,1,14,2,13,3,12,9,8,7],[12,5,11,6,1,7,2,14,3,13,4,10,9,8],[5,12,6,1,7,2,8,3,14,4,13,11,10,9],[13,6,12,7,2,8,3,9,4,14,5,1,11,10],[6,13,7,12,8,3,9,4,10,5,14,2,1,11],[14,7,13,8,12,9,4,10,5,11,6,3,2,1],[1,2,3,4,5,6,7,8,9,10,11,14,12,13],[11,1,2,3,4,5,6,7,8,9,10,13,14,12],[10,11,1,2,3,4,5,6,7,8,9,12,13,14]],[[14,13,3,9,4,10,5,11,6,12,7,1,8,2],[2,14,13,4,10,5,11,6,12,7,1,8,9,3],[9,3,14,13,5,11,6,12,7,1,8,2,10,4],[3,10,4,14,13,6,12,7,1,8,2,9,11,5],[10,4,11,5,14,13,7,1,8,2,9,3,12,6],[4,11,5,12,6,14,13,8,2,9,3,10,1,7],[11,5,12,6,1,7,14,13,9,3,10,4,2,8],[5,12,6,1,7,2,8,14,13,10,4,11,3,9],[12,6,1,7,2,8,3,9,14,13,11,5,4,10],[6,1,7,2,8,3,9,4,10,14,13,12,5,11],[1,7,2,8,3,9,4,10,5,11,14,13,6,12],[13,2,8,3,9,4,10,5,11,6,12,14,7,1],[8,9,10,11,12,1,2,3,4,5,6,7,14,13],[7,8,9,10,11,12,1,2,3,4,5,6,13,14]],[[1,4,2,7,3,11,8,13,5,14,6,9,10,12],[10,2,5,3,8,4,12,9,14,6,13,7,1,11],[8,11,3,6,4,9,5,1,10,13,7,14,12,2],[13,9,12,4,7,5,10,6,2,11,14,8,3,1],[9,14,10,1,5,8,6,11,7,3,12,13,2,4],[14,10,13,11,2,6,9,7,12,8,4,1,5,3],[2,13,11,14,12,3,7,10,8,1,9,5,4,6],[6,3,14,12,13,1,4,8,11,9,2,10,7,5],[11,7,4,13,1,14,2,5,9,12,10,3,6,8],[4,12,8,5,14,2,13,3,6,10,1,11,9,7],[12,5,1,9,6,13,3,14,4,7,11,2,8,10],[3,1,6,2,10,7,14,4,13,5,8,12,11,9],[5,6,7,8,9,10,11,12,1,2,3,4,14,13],[7,8,9,10,11,12,1,2,3,4,5,6,13,14]],[[8,14,9,3,10,4,11,5,12,6,13,7,1,2],[2,9,14,10,4,11,5,12,6,13,7,1,8,3],[9,3,10,14,11,5,12,6,13,7,1,8,2,4],[3,10,4,11,14,12,6,13,7,1,8,2,9,5],[10,4,11,5,12,14,13,7,1,8,2,9,3,6],[4,11,5,12,6,13,14,1,8,2,9,3,10,7],[11,5,12,6,13,7,1,14,2,9,3,10,4,8],[5,12,6,13,7,1,8,2,14,3,10,4,11,9],[12,6,13,7,1,8,2,9,3,14,4,11,5,10],[6,13,7,1,8,2,9,3,10,4,14,5,12,11],[13,7,1,8,2,9,3,10,4,11,5,14,6,12],[7,1,8,2,9,3,10,4,11,5,12,6,14,13],[14,8,2,9,3,10,4,11,5,12,6,13,7,1],[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]]; AutRep[15]:=[[(),(),()],[(),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)],[(),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2,3),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3)(4,5,6),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3)(4,5,6),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5)(6,7)(8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5)(6,7,8),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5)(6,7,8)(9,10,11),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5)(6,7,8)(9,10,11)(12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5)(6,7,8,9,10)(11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12)],[(1,2,3,4,5,6,7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)]]; LatinRep[15]:=[[[7,13,8,15,4,11,5,9,1,2,14,6,12,3,10],[2,8,13,3,6,5,15,12,9,1,7,11,10,4,14],[3,14,2,6,5,13,7,10,12,8,4,1,11,9,15],[10,2,7,1,14,9,13,4,11,15,3,12,5,8,6],[12,9,6,7,1,15,2,11,4,3,10,5,8,14,13],[9,10,1,5,8,12,4,3,6,14,2,7,13,15,11],[13,11,3,8,12,2,10,14,5,7,6,15,4,1,9],[14,5,9,10,2,6,12,1,8,4,11,3,15,13,7],[1,12,14,11,13,3,9,15,2,10,5,4,6,7,8],[6,1,5,4,9,14,8,13,3,11,15,10,7,2,12],[15,6,12,9,10,4,14,5,7,13,1,8,2,11,3],[5,15,10,2,7,1,11,8,14,6,9,13,3,12,4],[11,7,4,12,15,10,3,6,13,9,8,2,14,5,1],[4,3,15,13,11,8,1,7,10,5,12,14,9,6,2],[8,4,11,14,3,7,6,2,15,12,13,9,1,10,5]],[[4,5,6,1,2,3,7,8,9,13,14,15,10,11,12],[6,4,5,2,3,1,14,15,13,10,11,12,9,7,8],[1,2,3,7,8,9,10,11,12,15,13,14,4,5,6],[12,10,11,4,5,6,13,14,15,8,9,7,3,1,2],[5,6,4,10,11,12,15,13,14,1,2,3,7,8,9],[7,8,9,3,1,2,6,4,5,14,15,13,11,12,10],[10,11,12,13,14,15,1,2,3,5,6,4,8,9,7],[14,15,13,8,9,7,2,3,1,4,5,6,12,10,11],[13,14,15,5,6,4,8,9,7,12,10,11,1,2,3],[2,3,1,15,13,14,9,7,8,11,12,10,5,6,4],[3,1,2,14,15,13,11,12,10,7,8,9,6,4,5],[15,13,14,6,4,5,12,10,11,9,7,8,2,3,1],[11,12,10,9,7,8,4,5,6,2,3,1,13,14,15],[8,9,7,11,12,10,3,1,2,6,4,5,14,15,13],[9,7,8,12,10,11,5,6,4,3,1,2,15,13,14]],[[1,2,3,4,5,11,12,13,14,15,6,7,8,9,10],[15,11,12,13,14,6,7,8,9,10,2,3,4,5,1],[6,7,8,9,10,1,2,3,4,5,11,12,13,14,15],[11,12,13,14,15,7,8,9,10,6,1,2,3,4,5],[10,6,7,8,9,2,3,4,5,1,12,13,14,15,11],[2,3,4,5,1,15,11,12,13,14,7,8,9,10,6],[7,8,9,10,6,12,13,14,15,11,3,4,5,1,2],[12,13,14,15,11,3,4,5,1,2,8,9,10,6,7],[3,4,5,1,2,8,9,10,6,7,13,14,15,11,12],[8,9,10,6,7,13,14,15,11,12,4,5,1,2,3],[13,14,15,11,12,4,5,1,2,3,9,10,6,7,8],[4,5,1,2,3,9,10,6,7,8,14,15,11,12,13],[9,10,6,7,8,14,15,11,12,13,5,1,2,3,4],[14,15,11,12,13,5,1,2,3,4,10,6,7,8,9],[5,1,2,3,4,10,6,7,8,9,15,11,12,13,14]],[[3,4,5,6,7,8,9,10,11,12,13,14,15,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,1]],[[15,14,10,12,13,7,11,9,2,8,4,1,6,3,5],[14,15,12,10,8,13,9,11,1,7,3,2,5,4,6],[10,12,15,14,11,9,13,2,3,4,1,6,8,5,7],[12,10,14,15,9,11,1,13,4,3,2,5,7,6,8],[4,13,11,9,15,14,12,10,8,5,6,3,1,7,2],[13,3,9,11,14,15,10,12,7,6,5,4,2,8,1],[11,9,6,13,12,10,15,14,5,1,7,8,3,2,4],[9,11,13,5,10,12,14,15,6,2,8,7,4,1,3],[1,2,4,3,7,8,6,5,15,13,14,9,10,11,12],[7,8,3,4,6,5,2,1,10,15,13,14,11,12,9],[3,4,2,1,5,6,8,7,14,11,15,13,12,9,10],[2,1,5,6,4,3,7,8,13,14,12,15,9,10,11],[5,6,7,8,2,1,4,3,11,12,9,10,15,13,14],[6,5,8,7,1,2,3,4,12,9,10,11,13,14,15],[8,7,1,2,3,4,5,6,9,10,11,12,14,15,13]],[[15,8,4,12,14,10,6,11,13,1,2,3,5,7,9],[7,15,12,3,9,14,11,5,2,13,1,4,6,8,10],[3,13,15,10,6,12,14,1,8,11,4,5,7,9,2],[13,4,9,15,12,5,2,14,11,7,3,6,8,10,1],[11,10,5,13,15,1,8,12,14,3,6,7,9,2,4],[9,11,13,6,2,15,12,7,4,14,5,8,10,1,3],[5,14,11,1,7,13,15,3,10,12,8,9,2,4,6],[14,6,2,11,13,8,4,15,12,9,7,10,1,3,5],[12,1,7,14,11,3,9,13,15,5,10,2,4,6,8],[2,12,14,8,4,11,13,10,6,15,9,1,3,5,7],[4,3,6,5,8,7,10,9,1,2,13,14,15,11,12],[6,5,8,7,10,9,1,2,3,4,14,15,11,12,13],[8,7,10,9,1,2,3,4,5,6,15,11,12,13,14],[10,9,1,2,3,4,5,6,7,8,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,12,13,14,15,11]],[[1,11,3,13,5,15,7,12,9,14,2,8,4,10,6],[15,2,12,4,14,6,11,8,13,10,7,3,9,5,1],[2,12,4,14,6,11,8,13,10,15,1,7,3,9,5],[11,3,13,5,15,7,12,9,14,1,6,2,8,4,10],[3,13,5,15,7,12,9,14,1,11,4,10,6,2,8],[12,4,14,6,11,8,13,10,15,2,9,5,1,7,3],[4,14,6,11,8,13,10,15,2,12,3,9,5,1,7],[13,5,15,7,12,9,14,1,11,3,8,4,10,6,2],[5,15,7,12,9,14,1,11,3,13,10,6,2,8,4],[14,6,11,8,13,10,15,2,12,4,5,1,7,3,9],[6,7,8,9,10,1,2,3,4,5,11,12,13,14,15],[7,8,9,10,1,2,3,4,5,6,12,13,14,15,11],[8,9,10,1,2,3,4,5,6,7,13,14,15,11,12],[9,10,1,2,3,4,5,6,7,8,14,15,11,12,13],[10,1,2,3,4,5,6,7,8,9,15,11,12,13,14]],[[15,9,14,5,13,11,8,6,4,12,10,1,2,7,3],[10,15,6,14,12,13,5,7,11,3,2,9,1,8,4],[8,13,15,11,14,2,3,12,10,1,6,7,4,9,5],[13,7,12,15,1,14,11,4,2,9,8,5,3,10,6],[4,14,10,13,15,8,9,1,5,7,12,3,6,11,2],[14,3,13,9,7,15,2,10,8,6,4,11,5,12,1],[7,6,4,12,10,1,15,5,14,11,13,2,8,3,9],[5,8,11,3,2,9,6,15,12,14,1,13,7,4,10],[3,12,9,1,6,7,4,13,15,2,14,8,10,5,11],[11,4,2,10,8,5,13,3,1,15,7,14,9,6,12],[9,1,5,7,11,3,10,14,6,13,15,4,12,2,8],[2,10,8,6,4,12,14,9,13,5,3,15,11,1,7],[6,5,1,2,3,4,12,11,7,8,9,10,15,13,14],[12,11,7,8,9,10,1,2,3,4,5,6,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,14,15,13]],[[2,1,4,3,6,5,7,13,9,15,11,14,8,12,10],[6,3,2,5,4,1,15,8,14,10,13,12,11,9,7],[1,7,3,9,5,11,8,14,10,13,12,15,2,6,4],[12,2,8,4,10,6,13,9,15,11,14,7,5,3,1],[4,8,6,10,2,12,14,1,13,3,15,5,7,11,9],[7,5,9,1,11,3,6,15,2,14,4,13,10,8,12],[8,13,10,15,12,14,1,4,3,6,5,2,9,7,11],[15,9,14,11,13,7,3,2,5,4,1,6,12,10,8],[13,4,15,6,14,2,9,12,11,8,7,10,1,5,3],[3,14,5,13,1,15,11,10,7,12,9,8,4,2,6],[5,15,1,14,3,13,10,7,12,9,8,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,5,12,14,15,13],[6,7,8,9,10,11,12,1,2,3,4,5,13,14,15]],[[13,8,14,9,15,10,4,11,5,12,6,1,7,3,2],[2,14,9,15,10,13,11,5,12,6,1,7,3,8,4],[8,3,15,10,13,11,14,12,6,1,7,2,5,4,9],[3,9,4,13,11,14,12,15,1,7,2,8,10,6,5],[9,4,10,5,14,12,15,1,13,2,8,3,6,11,7],[4,10,5,11,6,15,1,13,2,14,3,9,8,7,12],[10,5,11,6,12,7,13,2,14,3,15,4,1,9,8],[5,11,6,12,7,1,8,14,3,15,4,13,9,2,10],[14,6,12,7,1,8,2,9,15,4,13,5,11,10,3],[6,15,7,1,8,2,9,3,10,13,5,14,4,12,11],[15,7,13,8,2,9,3,10,4,11,14,6,12,5,1],[7,13,8,14,9,3,10,4,11,5,12,15,2,1,6],[12,2,1,3,5,4,6,8,7,9,11,10,15,14,13],[11,1,3,2,4,6,5,7,9,8,10,12,14,13,15],[1,12,2,4,3,5,7,6,8,10,9,11,13,15,14]],[[8,15,9,14,10,4,11,5,12,6,13,7,1,3,2],[2,9,15,10,14,11,5,12,6,13,7,1,8,4,3],[9,3,10,15,11,14,12,6,13,7,1,8,2,5,4],[3,10,4,11,15,12,14,13,7,1,8,2,9,6,5],[10,4,11,5,12,15,13,14,1,8,2,9,3,7,6],[4,11,5,12,6,13,15,1,14,2,9,3,10,8,7],[11,5,12,6,13,7,1,15,2,14,3,10,4,9,8],[5,12,6,13,7,1,8,2,15,3,14,4,11,10,9],[12,6,13,7,1,8,2,9,3,15,4,14,5,11,10],[6,13,7,1,8,2,9,3,10,4,15,5,14,12,11],[14,7,1,8,2,9,3,10,4,11,5,15,6,13,12],[7,14,8,2,9,3,10,4,11,5,12,6,15,1,13],[15,8,14,9,3,10,4,11,5,12,6,13,7,2,1],[1,2,3,4,5,6,7,8,9,10,11,12,13,15,14],[13,1,2,3,4,5,6,7,8,9,10,11,12,14,15]],[[15,9,2,10,3,11,4,12,5,13,6,14,7,1,8],[2,15,10,3,11,4,12,5,13,6,14,7,1,8,9],[9,3,15,11,4,12,5,13,6,14,7,1,8,2,10],[3,10,4,15,12,5,13,6,14,7,1,8,2,9,11],[10,4,11,5,15,13,6,14,7,1,8,2,9,3,12],[4,11,5,12,6,15,14,7,1,8,2,9,3,10,13],[11,5,12,6,13,7,15,1,8,2,9,3,10,4,14],[5,12,6,13,7,14,8,15,2,9,3,10,4,11,1],[12,6,13,7,14,8,1,9,15,3,10,4,11,5,2],[6,13,7,14,8,1,9,2,10,15,4,11,5,12,3],[13,7,14,8,1,9,2,10,3,11,15,5,12,6,4],[7,14,8,1,9,2,10,3,11,4,12,15,6,13,5],[14,8,1,9,2,10,3,11,4,12,5,13,15,7,6],[8,1,9,2,10,3,11,4,12,5,13,6,14,15,7],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],[[9,2,10,3,11,4,12,5,13,6,14,7,15,8,1],[2,10,3,11,4,12,5,13,6,14,7,15,8,1,9],[10,3,11,4,12,5,13,6,14,7,15,8,1,9,2],[3,11,4,12,5,13,6,14,7,15,8,1,9,2,10],[11,4,12,5,13,6,14,7,15,8,1,9,2,10,3],[4,12,5,13,6,14,7,15,8,1,9,2,10,3,11],[12,5,13,6,14,7,15,8,1,9,2,10,3,11,4],[5,13,6,14,7,15,8,1,9,2,10,3,11,4,12],[13,6,14,7,15,8,1,9,2,10,3,11,4,12,5],[6,14,7,15,8,1,9,2,10,3,11,4,12,5,13],[14,7,15,8,1,9,2,10,3,11,4,12,5,13,6],[7,15,8,1,9,2,10,3,11,4,12,5,13,6,14],[15,8,1,9,2,10,3,11,4,12,5,13,6,14,7],[8,1,9,2,10,3,11,4,12,5,13,6,14,7,15],[1,9,2,10,3,11,4,12,5,13,6,14,7,15,8]]]; AutRep[16]:=[[(),(),()],[(),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4)(5,6),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)],[(1,2)(3,4)(5,6)(7,8),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5)(6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5)(6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15)],[(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9)(10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10)(11,12),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)]]; LatinRep[16]:=[[[1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,9],[2,3,4,5,6,7,8,1,9,10,11,12,13,14,15,16],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[16,9,10,11,12,13,14,15,2,3,4,5,6,7,8,1],[10,11,12,13,14,15,16,9,3,4,5,6,7,8,1,2],[3,4,5,6,7,8,1,2,11,12,13,14,15,16,9,10],[11,12,13,14,15,16,9,10,4,5,6,7,8,1,2,3],[4,5,6,7,8,1,2,3,12,13,14,15,16,9,10,11],[12,13,14,15,16,9,10,11,5,6,7,8,1,2,3,4],[5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12],[13,14,15,16,9,10,11,12,6,7,8,1,2,3,4,5],[6,7,8,1,2,3,4,5,14,15,16,9,10,11,12,13],[14,15,16,9,10,11,12,13,7,8,1,2,3,4,5,6],[7,8,1,2,3,4,5,6,15,16,9,10,11,12,13,14],[15,16,9,10,11,12,13,14,8,1,2,3,4,5,6,7],[8,1,2,3,4,5,6,7,16,9,10,11,12,13,14,15]],[[1,2,4,3,5,6,8,7,9,10,12,11,13,14,16,15],[2,1,3,4,6,5,7,8,10,9,11,12,14,13,15,16],[3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14],[4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13],[5,6,8,7,1,2,4,3,13,14,16,15,9,10,12,11],[6,5,7,8,2,1,3,4,14,13,15,16,10,9,11,12],[7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10],[8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[11,12,10,9,15,16,14,13,3,4,2,1,7,8,6,5],[10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7],[12,11,9,10,16,15,13,14,4,3,1,2,8,7,5,6],[13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4],[15,16,14,13,11,12,10,9,7,8,6,5,3,4,2,1],[14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3],[16,15,13,14,12,11,9,10,8,7,5,6,4,3,1,2]],[[1,2,3,4,6,7,8,5,9,10,11,12,14,15,16,13],[2,3,4,1,5,6,7,8,10,11,12,9,13,14,15,16],[5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12],[8,5,6,7,2,3,4,1,16,13,14,15,10,11,12,9],[6,7,8,5,4,1,2,3,14,15,16,13,12,9,10,11],[7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10],[3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14],[4,1,2,3,8,5,6,7,12,9,10,11,16,13,14,15],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[13,14,15,16,10,11,12,9,5,6,7,8,2,3,4,1],[10,11,12,9,14,15,16,13,2,3,4,1,6,7,8,5],[14,15,16,13,11,12,9,10,6,7,8,5,3,4,1,2],[11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6],[15,16,13,14,12,9,10,11,7,8,5,6,4,1,2,3],[12,9,10,11,16,13,14,15,4,1,2,3,8,5,6,7],[16,13,14,15,9,10,11,12,8,5,6,7,1,2,3,4]],[[1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,9],[2,3,4,5,6,7,8,1,9,10,11,12,13,14,15,16],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[16,9,10,11,12,13,14,15,2,3,4,5,6,7,8,1],[10,11,12,13,14,15,16,9,3,4,5,6,7,8,1,2],[3,4,5,6,7,8,1,2,11,12,13,14,15,16,9,10],[11,12,13,14,15,16,9,10,4,5,6,7,8,1,2,3],[4,5,6,7,8,1,2,3,12,13,14,15,16,9,10,11],[12,13,14,15,16,9,10,11,5,6,7,8,1,2,3,4],[5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12],[13,14,15,16,9,10,11,12,6,7,8,1,2,3,4,5],[6,7,8,1,2,3,4,5,14,15,16,9,10,11,12,13],[14,15,16,9,10,11,12,13,7,8,1,2,3,4,5,6],[7,8,1,2,3,4,5,6,15,16,9,10,11,12,13,14],[15,16,9,10,11,12,13,14,8,1,2,3,4,5,6,7],[8,1,2,3,4,5,6,7,16,9,10,11,12,13,14,15]],[[3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1]],[[1,3,2,4,5,7,6,8,9,11,10,12,13,15,14,16],[4,2,3,1,8,6,7,5,12,10,11,9,16,14,15,13],[2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15],[3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14],[5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12],[6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11],[7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10],[8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7],[11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6],[12,11,10,9,16,15,14,13,4,3,2,1,8,7,6,5],[13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4],[14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3],[15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2],[16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]],[[1,4,3,2,5,8,7,6,9,12,11,10,13,16,15,14],[3,2,1,4,7,6,5,8,11,10,9,12,15,14,13,16],[2,3,4,1,6,7,8,5,10,11,12,9,14,15,16,13],[4,1,2,3,8,5,6,7,12,9,10,11,16,13,14,15],[5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12],[6,7,8,5,2,3,4,1,14,15,16,13,10,11,12,9],[7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10],[8,5,6,7,4,1,2,3,16,13,14,15,12,9,10,11],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[10,11,12,9,14,15,16,13,2,3,4,1,6,7,8,5],[11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6],[12,9,10,11,16,13,14,15,4,1,2,3,8,5,6,7],[13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4],[14,15,16,13,10,11,12,9,6,7,8,5,2,3,4,1],[15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2],[16,13,14,15,12,9,10,11,8,5,6,7,4,1,2,3]],[[1,4,3,6,5,8,7,2,9,12,11,14,13,16,15,10],[3,2,5,4,7,6,1,8,11,10,13,12,15,14,9,16],[2,3,4,5,6,7,8,1,10,11,12,13,14,15,16,9],[4,5,6,7,8,1,2,3,12,13,14,15,16,9,10,11],[5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12],[6,7,8,1,2,3,4,5,14,15,16,9,10,11,12,13],[7,8,1,2,3,4,5,6,15,16,9,10,11,12,13,14],[8,1,2,3,4,5,6,7,16,9,10,11,12,13,14,15],[9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8],[10,11,12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,10,11,12],[6,14,7,1,8,2,9,3,10,4,16,5,15,11,12,13],[15,7,14,8,2,9,3,10,4,11,5,16,6,12,13,1],[7,15,8,14,9,3,10,4,11,5,12,6,16,13,1,2],[16,8,15,9,14,10,4,11,5,12,6,13,7,1,2,3],[12,13,1,2,3,4,5,6,7,8,9,10,11,16,14,15],[13,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,14]],[[15,9,2,10,3,11,4,12,5,13,6,14,7,16,8,1],[16,15,10,3,11,4,12,5,13,6,14,7,1,8,9,2],[9,16,15,11,4,12,5,13,6,14,7,1,8,2,10,3],[3,10,16,15,12,5,13,6,14,7,1,8,2,9,11,4],[10,4,11,16,15,13,6,14,7,1,8,2,9,3,12,5],[4,11,5,12,16,15,14,7,1,8,2,9,3,10,13,6],[11,5,12,6,13,16,15,1,8,2,9,3,10,4,14,7],[5,12,6,13,7,14,16,15,2,9,3,10,4,11,1,8],[12,6,13,7,14,8,1,16,15,3,10,4,11,5,2,9],[6,13,7,14,8,1,9,2,16,15,4,11,5,12,3,10],[13,7,14,8,1,9,2,10,3,16,15,5,12,6,4,11],[7,14,8,1,9,2,10,3,11,4,16,15,6,13,5,12],[14,8,1,9,2,10,3,11,4,12,5,16,15,7,6,13],[8,1,9,2,10,3,11,4,12,5,13,6,16,15,7,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,1,15,16]],[[1,3,2,6,8,4,11,13,15,5,16,9,7,10,12,14],[11,2,4,3,7,9,5,12,14,16,6,15,10,8,1,13],[9,12,3,5,4,8,10,6,13,1,15,7,16,11,14,2],[12,10,13,4,6,5,9,11,7,14,2,16,8,15,3,1],[16,13,11,14,5,7,6,10,12,8,1,3,15,9,2,4],[10,15,14,12,1,6,8,7,11,13,9,2,4,16,5,3],[15,11,16,1,13,2,7,9,8,12,14,10,3,5,4,6],[6,16,12,15,2,14,3,8,10,9,13,1,11,4,7,5],[5,7,15,13,16,3,1,4,9,11,10,14,2,12,6,8],[13,6,8,16,14,15,4,2,5,10,12,11,1,3,9,7],[4,14,7,9,15,1,16,5,3,6,11,13,12,2,8,10],[3,5,1,8,10,16,2,15,6,4,7,12,14,13,11,9],[14,4,6,2,9,11,15,3,16,7,5,8,13,1,10,12],[2,1,5,7,3,10,12,16,4,15,8,6,9,14,13,11],[7,8,9,10,11,12,13,14,1,2,3,4,5,6,15,16],[8,9,10,11,12,13,14,1,2,3,4,5,6,7,16,15]],[[9,16,10,3,11,4,12,5,13,6,14,7,15,8,1,2],[2,10,16,11,4,12,5,13,6,14,7,15,8,1,9,3],[10,3,11,16,12,5,13,6,14,7,15,8,1,9,2,4],[3,11,4,12,16,13,6,14,7,15,8,1,9,2,10,5],[11,4,12,5,13,16,14,7,15,8,1,9,2,10,3,6],[4,12,5,13,6,14,16,15,8,1,9,2,10,3,11,7],[12,5,13,6,14,7,15,16,1,9,2,10,3,11,4,8],[5,13,6,14,7,15,8,1,16,2,10,3,11,4,12,9],[13,6,14,7,15,8,1,9,2,16,3,11,4,12,5,10],[6,14,7,15,8,1,9,2,10,3,16,4,12,5,13,11],[14,7,15,8,1,9,2,10,3,11,4,16,5,13,6,12],[7,15,8,1,9,2,10,3,11,4,12,5,16,6,14,13],[15,8,1,9,2,10,3,11,4,12,5,13,6,16,7,14],[8,1,9,2,10,3,11,4,12,5,13,6,14,7,16,15],[16,9,2,10,3,11,4,12,5,13,6,14,7,15,8,1],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]]; AutRep[17]:=[[(),(),()],[(),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)],[(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10),(1,2)(3,4)(5,6)(7,8)(9,10)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)],[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16),(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)],[(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9),(1,2,3)(4,5,6)(7,8,9)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12),(1,2,3)(4,5,6)(7,8,9)(10,11,12)],[(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15),(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12),(1,2,3,4)(5,6,7,8)(9,10,11,12)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14)(15,16)],[(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16),(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)],[(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10),(1,2,3,4,5)(6,7,8,9,10)],[(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15),(1,2,3,4,5,6)(7,8,9)(10,11)(12,13)(14,15)],[(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9)(10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12),(1,2,3,4,5,6)(7,8,9,10,11,12)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14),(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9)],[(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9)(10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10),(1,2,3,4,5,6,7,8,9,10)],[(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)],[(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10)(11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11),(1,2,3,4,5,6,7,8,9,10,11)],[(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12),(1,2,3,4,5,6,7,8,9,10,11,12)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13),(1,2,3,4,5,6,7,8,9,10,11,12,13)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)],[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17),(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)]]; LatinRep[17]:=[[[12,15,4,9,17,10,3,7,16,14,8,5,13,1,11,2,6],[17,13,14,1,9,12,4,2,11,15,7,3,10,5,8,6,16],[10,12,1,13,14,15,8,16,3,2,11,9,4,6,17,5,7],[1,14,3,15,16,7,11,13,9,6,17,4,5,10,2,12,8],[4,16,12,3,8,2,14,10,17,11,5,6,15,9,1,7,13],[8,17,6,7,13,16,2,4,15,5,12,10,1,11,3,14,9],[11,4,15,16,10,13,7,12,14,17,6,1,3,8,5,9,2],[6,9,5,11,1,8,13,17,7,16,3,14,12,2,15,4,10],[9,3,13,4,2,6,17,15,5,10,16,11,14,12,7,8,1],[13,10,7,6,15,1,16,5,8,3,9,2,11,14,12,17,4],[15,6,16,17,12,3,5,1,2,8,10,7,9,4,13,11,14],[3,8,2,12,5,11,10,14,4,13,15,16,6,7,9,1,17],[14,2,11,5,6,17,1,9,12,7,4,13,8,3,10,16,15],[5,11,9,2,3,14,6,8,10,4,1,17,7,15,16,13,12],[16,7,17,8,11,4,15,3,1,9,14,12,2,13,6,10,5],[2,1,10,14,7,5,9,11,6,12,13,8,16,17,4,15,3],[7,5,8,10,4,9,12,6,13,1,2,15,17,16,14,3,11]],[[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,1]],[[11,8,4,14,12,17,6,15,13,16,2,7,3,9,5,10,1],[7,11,14,3,17,12,15,5,16,13,1,8,4,10,6,9,2],[16,14,12,10,6,15,13,17,8,11,7,4,9,5,2,1,3],[14,16,9,12,15,5,17,13,11,7,8,3,10,6,1,2,4],[12,10,16,15,13,1,8,11,14,17,4,9,6,2,7,3,5],[9,12,15,16,2,13,11,7,17,14,3,10,5,1,8,4,6],[17,15,13,1,16,11,14,3,10,12,9,6,2,8,4,5,7],[15,17,2,13,11,16,4,14,12,9,10,5,1,7,3,6,8],[13,1,17,11,14,3,16,12,15,5,6,2,8,4,10,7,9],[2,13,11,17,4,14,12,16,6,15,5,1,7,3,9,8,10],[8,7,3,4,10,9,5,6,1,2,14,17,15,16,11,12,13],[4,3,10,9,5,6,1,2,7,8,12,15,17,11,16,13,14],[10,9,6,5,1,2,7,8,3,4,16,13,11,17,12,14,15],[6,5,1,2,8,7,3,4,9,10,13,16,14,12,17,15,11],[1,2,8,7,3,4,10,9,5,6,17,14,16,15,13,11,12],[3,4,5,6,7,8,9,10,2,1,15,11,12,13,14,16,17],[5,6,7,8,9,10,2,1,4,3,11,12,13,14,15,17,16]],[[13,7,4,10,14,9,6,17,15,11,8,16,2,5,3,12,1],[8,13,9,3,10,14,17,5,12,15,16,7,1,6,4,11,2],[16,10,14,9,6,12,15,11,8,17,13,2,5,4,7,1,3],[9,16,10,14,11,5,12,15,17,7,1,13,6,3,8,2,4],[4,14,16,12,15,11,8,1,13,2,10,17,9,7,6,3,5],[14,3,11,16,12,15,2,7,1,13,17,9,10,8,5,4,6],[17,12,6,15,16,1,13,2,10,3,14,4,8,11,9,5,7],[11,17,15,5,2,16,1,13,4,9,3,14,7,12,10,6,8],[6,15,17,1,8,13,16,3,14,4,12,5,11,10,2,7,9],[15,5,2,17,13,7,4,16,3,14,6,11,12,9,1,8,10],[7,1,8,13,17,3,10,14,16,5,15,6,4,2,12,9,11],[2,8,13,7,4,17,14,9,6,16,5,15,3,1,11,10,12],[12,11,3,4,1,2,5,6,9,10,7,8,16,17,13,14,15],[10,9,1,2,5,6,3,4,7,8,11,12,14,16,17,15,13],[1,2,12,11,3,4,7,8,5,6,9,10,17,15,16,13,14],[3,4,5,6,7,8,9,10,11,12,2,1,15,13,14,16,17],[5,6,7,8,9,10,11,12,2,1,4,3,13,14,15,17,16]],[[15,8,11,13,16,10,12,14,17,1,2,6,4,9,7,5,3],[7,15,14,12,9,16,13,11,2,17,5,1,10,3,8,6,4],[3,17,15,10,12,14,16,1,11,13,8,4,2,6,9,7,5],[17,4,9,15,13,11,2,16,14,12,3,7,5,1,10,8,6],[14,12,5,17,15,1,11,13,16,3,6,10,8,4,2,9,7],[11,13,17,6,2,15,14,12,4,16,9,5,3,7,1,10,8],[5,16,13,11,7,17,15,3,12,14,1,8,6,10,4,2,9],[16,6,12,14,17,8,4,15,13,11,7,2,9,5,3,1,10],[13,11,7,16,14,12,9,17,15,5,10,3,1,8,6,4,2],[12,14,16,8,11,13,17,10,6,15,4,9,7,2,5,3,1],[2,5,8,3,6,9,1,7,10,4,17,16,15,11,13,12,14],[6,1,4,7,10,5,8,2,3,9,16,17,12,15,14,11,13],[4,10,2,5,8,3,6,9,1,7,13,15,17,16,12,14,11],[9,3,6,1,4,7,10,5,8,2,15,14,16,17,11,13,12],[8,7,10,9,1,2,3,4,5,6,12,11,14,13,17,15,16],[1,2,3,4,5,6,7,8,9,10,14,13,11,12,15,16,17],[10,9,1,2,3,4,5,6,7,8,11,12,13,14,16,17,15]],[[17,5,9,13,3,7,11,15,10,14,4,8,12,16,6,1,2],[6,17,14,10,8,4,16,12,13,9,7,3,15,11,2,5,1],[10,13,17,7,11,15,5,2,3,8,12,16,6,1,14,9,4],[14,9,8,17,16,12,1,6,7,4,15,11,2,5,10,13,3],[4,7,12,15,17,2,13,10,11,16,5,1,14,9,8,3,6],[8,3,16,11,1,17,9,14,15,12,2,6,10,13,4,7,5],[12,15,6,2,14,10,17,4,5,1,13,9,7,3,16,11,8],[16,11,1,5,9,13,3,17,2,6,10,14,4,8,12,15,7],[9,14,4,8,12,16,6,1,17,13,3,7,11,15,5,2,10],[13,10,7,3,15,11,2,5,14,17,8,4,16,12,1,6,9],[3,8,11,16,6,1,14,9,4,7,17,15,5,2,13,10,12],[7,4,15,12,2,5,10,13,8,3,16,17,1,6,9,14,11],[11,16,5,1,13,9,8,3,12,15,6,2,17,10,7,4,14],[15,12,2,6,10,14,4,7,16,11,1,5,9,17,3,8,13],[5,1,13,9,7,3,15,11,6,2,14,10,8,4,17,12,16],[2,6,10,14,4,8,12,16,1,5,9,13,3,7,11,17,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],[[13,16,10,17,11,14,12,15,1,4,7,2,5,8,3,6,9],[10,13,16,14,17,11,2,12,15,5,8,3,6,9,1,4,7],[16,10,13,11,14,17,15,3,12,6,9,1,4,7,2,5,8],[4,12,15,13,16,10,17,11,14,7,2,5,8,3,6,9,1],[15,5,12,10,13,16,14,17,11,8,3,6,9,1,4,7,2],[12,15,6,16,10,13,11,14,17,9,1,4,7,2,5,8,3],[14,17,11,7,12,15,13,16,10,2,5,8,3,6,9,1,4],[11,14,17,15,8,12,10,13,16,3,6,9,1,4,7,2,5],[17,11,14,12,15,9,16,10,13,1,4,7,2,5,8,3,6],[7,8,9,2,3,1,5,6,4,12,13,14,15,16,17,10,11],[2,3,1,5,6,4,8,9,7,13,14,15,16,17,10,11,12],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7,14,16,1,15,2,9,3,10,13,7,8],[11,5,12,6,13,7,17,1,16,2,15,3,10,4,14,8,9],[5,12,6,13,7,14,8,17,2,16,3,15,4,11,1,9,10],[12,6,13,7,14,8,1,9,17,3,16,4,15,5,2,10,11],[6,13,7,14,8,1,9,2,10,17,4,16,5,15,3,11,12],[15,7,14,8,1,9,2,10,3,11,17,5,16,6,4,12,13],[7,15,8,1,9,2,10,3,11,4,12,17,6,16,5,13,14],[16,8,15,9,2,10,3,11,4,12,5,13,17,7,6,14,1],[8,16,9,15,10,3,11,4,12,5,13,6,14,17,7,1,2],[13,14,1,2,3,4,5,6,7,8,9,10,11,12,17,15,16],[14,1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,15]],[[17,15,2,16,3,11,4,12,5,13,6,14,7,1,10,8,9],[2,17,16,3,15,4,12,5,13,6,14,7,1,8,9,11,10],[9,3,17,15,4,16,5,13,6,14,7,1,8,2,12,10,11],[3,10,4,17,16,5,15,6,14,7,1,8,2,9,11,13,12],[10,4,11,5,17,15,6,16,7,1,8,2,9,3,14,12,13],[4,11,5,12,6,17,16,7,15,8,2,9,3,10,13,1,14],[11,5,12,6,13,7,17,15,8,16,9,3,10,4,2,14,1],[5,12,6,13,7,14,8,17,16,9,15,10,4,11,1,3,2],[12,6,13,7,14,8,1,9,17,15,10,16,11,5,4,2,3],[6,13,7,14,8,1,9,2,10,17,16,11,15,12,3,5,4],[13,7,14,8,1,9,2,10,3,11,17,15,12,16,6,4,5],[15,14,8,1,9,2,10,3,11,4,12,17,16,13,5,7,6],[14,16,1,9,2,10,3,11,4,12,5,13,17,15,8,6,7],[16,1,15,2,10,3,11,4,12,5,13,6,14,17,7,9,8],[1,8,3,10,5,12,7,14,9,2,11,4,13,6,17,15,16],[7,2,9,4,11,6,13,8,1,10,3,12,5,14,16,17,15],[8,9,10,11,12,13,14,1,2,3,4,5,6,7,15,16,17]],[[9,16,10,3,11,4,12,5,13,6,14,7,15,8,17,2,1],[17,10,16,11,4,12,5,13,6,14,7,15,8,1,9,3,2],[10,17,11,16,12,5,13,6,14,7,15,8,1,9,2,4,3],[3,11,17,12,16,13,6,14,7,15,8,1,9,2,10,5,4],[11,4,12,17,13,16,14,7,15,8,1,9,2,10,3,6,5],[4,12,5,13,17,14,16,15,8,1,9,2,10,3,11,7,6],[12,5,13,6,14,17,15,16,1,9,2,10,3,11,4,8,7],[5,13,6,14,7,15,17,1,16,2,10,3,11,4,12,9,8],[13,6,14,7,15,8,1,17,2,16,3,11,4,12,5,10,9],[6,14,7,15,8,1,9,2,17,3,16,4,12,5,13,11,10],[14,7,15,8,1,9,2,10,3,17,4,16,5,13,6,12,11],[7,15,8,1,9,2,10,3,11,4,17,5,16,6,14,13,12],[15,8,1,9,2,10,3,11,4,12,5,17,6,16,7,14,13],[8,1,9,2,10,3,11,4,12,5,13,6,17,7,16,15,14],[16,9,2,10,3,11,4,12,5,13,6,14,7,17,8,1,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,1,16,17]],[[17,10,2,11,3,12,4,13,5,14,6,15,7,16,8,1,9],[2,17,11,3,12,4,13,5,14,6,15,7,16,8,1,9,10],[10,3,17,12,4,13,5,14,6,15,7,16,8,1,9,2,11],[3,11,4,17,13,5,14,6,15,7,16,8,1,9,2,10,12],[11,4,12,5,17,14,6,15,7,16,8,1,9,2,10,3,13],[4,12,5,13,6,17,15,7,16,8,1,9,2,10,3,11,14],[12,5,13,6,14,7,17,16,8,1,9,2,10,3,11,4,15],[5,13,6,14,7,15,8,17,1,9,2,10,3,11,4,12,16],[13,6,14,7,15,8,16,9,17,2,10,3,11,4,12,5,1],[6,14,7,15,8,16,9,1,10,17,3,11,4,12,5,13,2],[14,7,15,8,16,9,1,10,2,11,17,4,12,5,13,6,3],[7,15,8,16,9,1,10,2,11,3,12,17,5,13,6,14,4],[15,8,16,9,1,10,2,11,3,12,4,13,17,6,14,7,5],[8,16,9,1,10,2,11,3,12,4,13,5,14,17,7,15,6],[16,9,1,10,2,11,3,12,4,13,5,14,6,15,17,8,7],[9,1,10,2,11,3,12,4,13,5,14,6,15,7,16,17,8],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],[[10,2,11,3,12,4,13,5,14,6,15,7,16,8,17,9,1],[2,11,3,12,4,13,5,14,6,15,7,16,8,17,9,1,10],[11,3,12,4,13,5,14,6,15,7,16,8,17,9,1,10,2],[3,12,4,13,5,14,6,15,7,16,8,17,9,1,10,2,11],[12,4,13,5,14,6,15,7,16,8,17,9,1,10,2,11,3],[4,13,5,14,6,15,7,16,8,17,9,1,10,2,11,3,12],[13,5,14,6,15,7,16,8,17,9,1,10,2,11,3,12,4],[5,14,6,15,7,16,8,17,9,1,10,2,11,3,12,4,13],[14,6,15,7,16,8,17,9,1,10,2,11,3,12,4,13,5],[6,15,7,16,8,17,9,1,10,2,11,3,12,4,13,5,14],[15,7,16,8,17,9,1,10,2,11,3,12,4,13,5,14,6],[7,16,8,17,9,1,10,2,11,3,12,4,13,5,14,6,15],[16,8,17,9,1,10,2,11,3,12,4,13,5,14,6,15,7],[8,17,9,1,10,2,11,3,12,4,13,5,14,6,15,7,16],[17,9,1,10,2,11,3,12,4,13,5,14,6,15,7,16,8],[9,1,10,2,11,3,12,4,13,5,14,6,15,7,16,8,17],[1,10,2,11,3,12,4,13,5,14,6,15,7,16,8,17,9]]]; ################################################################### # # We begin with a list of functions that are used in the code # # # IsLatinSquare(L) # # Input: matrix L # Output: true if L is a Latin square (on any symbol set), otherwise false. # IsLatinSquare:=function(L) local L2; if(not IsMatrix(L)) then return false; fi; # checks if L is a matrix if(Size(L)<>Size(L[1])) then return false; fi; # checks if L is a square matrix if(not ForAll(L,r->Size(r)=Size(Set(r)))) then return false; fi; # checks if rows of L are Latin if(not ForAll(TransposedMat(L),r->Size(r)=Size(Set(r)))) then return false; fi; # checks if columns of L are Latin return true; end;; # # ConstructionFromOrthogonalArray(O,n) # # Input: orthogonal array O and order n # For each entry e=[r,c,s] in O assigns L[r][c]=s or 0 otherwise # Output: L # ConstructionFromOrthogonalArray:=function(O,n) local L,e; L:=List([1..n],i->List([1..n],j->0)); for e in O do L[e[1]][e[2]]:=e[3]; od; return L; end; # # CycleStructureToPermutation(P) # # Input: ordered partition P of natural number n # Output: permutation with cycle structure P # # For example: # # gap> CycleStructureToPermutation([3,5,2]); # (1,2,3)(4,5,6,7,8)(9,10) # CycleStructureToPermutation:=function(P) local l,i; l:=[]; for i in [1..Size(P)] do Append(l,List([1..P[i]],j->(j mod P[i])+1)+Sum([1..i-1],k->P[k])); od; return PermList(l); end;; # # GenerateOrthogonalArrayFromContourAndAutomorphism(O,alpha) # GenerateOrthogonalArrayFromContourAndAutotopism(O,alpha,beta,gamma) # # Input: partial orthogonal array O and permutations alpha, or alpha,beta,gamma # Output: generates the rest of the orthogonal array O from <(alpha,alpha,alpha)> # or <(alpha,beta,gamma)> # # Typically we input a contour as a partial orthogonal array O # from which we generate a Latin square that admits the automorphism (alpha,beta,gamma) # GenerateOrthogonalArrayFromContourAndAutotopism:=function(O,alpha,beta,gamma) local ONew,e,i,l; ONew:=[]; l:=Lcm(Order(alpha),Order(beta),Order(gamma)); for e in O do for i in [1..l] do Append(ONew,[[e[1]^(alpha^i),e[2]^(beta^i),e[3]^(gamma^i)]]); od; od; return Set(ONew); end;; GenerateOrthogonalArrayFromContourAndAutomorphism:=function(O,alpha) return GenerateOrthogonalArrayFromContourAndAutotopism(O,alpha,alpha,alpha); end;; # # PermuteRowsMatrix(L,perm) # PermuteColumnsMatrix(L,perm) # PermuteSymbolsMatrix(L,perm) # # Input: Matrix L and permutation perm # Output: Matrix formed by permuting the rows of L by perm # and similarly for columns and symbols # PermuteRowsMatrix:=function(L,perm) if(not IsMatrix(L)) then return fail; fi; if(IsSubset([1..Size(L)],ListPerm(perm))) then return List([1..Size(L)],i->L[i^Inverse(perm)]); else return fail; fi; end;; PermuteColumnsMatrix:=function(L,perm) if(not IsMatrix(L)) then return fail; fi; return TransposedMat(PermuteRowsMatrix(TransposedMat(L),perm)); end;; LatinSquareCMap:=function(c,perm) if(c=0) then return 0; else return c^perm; fi; end;; PermuteSymbolsMatrix:=function(L,perm) if(not IsMatrix(L)) then return fail; fi; return List(L,r->List(r,c->LatinSquareCMap(c,perm))); end;; # # DirectProductLatinSquares(L1,L2) # # Input: Two Latin squares L1 and L2 # Output: The direct product of L1 and L2 # DirectProductLatinSquares:=function(L1,L2) local r,c,M,n,a,b; a:=Size(L1); b:=Size(L2); n:=a*b; M:=[]; for r in [1..n] do M[r]:=[]; for c in [1..n] do M[r][c]:=L1[((r-1) mod a)+1][((c-1) mod a)+1]+L2[Int((r-1)/a)+1][Int((c-1)/a)+1]*a; od; od; return M-a; end;; # # IsotopicMatrix(L,alpha,beta,gamma) # # Input: Matrix L and three permutations alpha, beta, gamma. # Output: Matrix formed by permuting the # rows of L by alpha # columns of L by beta # symbols of L by gamma # IsotopicMatrix:=function(L,alpha,beta,gamma) if(not IsMatrix(L)) then return fail; fi; return PermuteSymbolsMatrix(PermuteColumnsMatrix(PermuteRowsMatrix(L,alpha),beta),gamma); end;; # # IsAutomorphismLatinSquare(L,alpha) # IsAutotopismLatinSquare(L,alpha,beta,gamma) # # Input: Matrix L and permutation alpha (or alpha, beta and gamma) # Output: returns true of L is a Latin square and alpha is an automorphism of L # (or (alpha,beta,gamma) is an autotopism of L) # IsAutomorphismLatinSquare:=function(L,alpha) if(not IsLatinSquare(L)) then return false; fi; return L=IsotopicMatrix(L,alpha,alpha,alpha); end; IsAutotopismLatinSquare:=function(L,alpha,beta,gamma) if(not IsLatinSquare(L)) then return false; fi; return L=IsotopicMatrix(L,alpha,beta,gamma); end; # # ExistsLatinSquareOneCycle(d,f) # ExistsLatinSquareTwoCycles(d1,d2,f) # ExistsLatinSquareThreeCycles(d1,d2,d3,f) # # Returns true if there exists a Latin square with automorphism alpha # where alpha has the cycle structure: # [d,1,1,...,1] # [d1,d2,1,1,...,1] # [d1,d2,d3,1,1,...,1] # with f 1-cycles (fixed points) # # Warning: assumes d1>=d2>=d3 # ExistsLatinSquareOneCycle:=function(d,f) return (f=0 and d mod 2=1) or (1<=f and d>=f); end;; ExistsLatinSquareTwoCycles:=function(d1,d2,f) if(d1d2 and d1 mod d2=0 and d1>=(d1+d2+f)/2 and d2>=f and (d2 mod 2=1 or f>0) or d1=(d2+d1+f)/2 and d1>=f and (d1 mod 2=1 or f>0); end;; ExistsLatinSquareThreeCycles:=function(d1,d2,d3,f) if(d1d2 and d2=d3) then if(not d1 mod d2=0) then return false; else return d1>=2*d2+f and f<=2*d2 and (not (d2 mod 2=0 and d1/d2 mod 2=1) or f>0); fi; elif(d1=d2 and d2>d3) then return d1 mod d3=0 and f<=d3 and (d3 mod 2=1 or f>0); else return d1=Lcm(d2,d3) and d3>=f and (d1 mod 2=1 or f>0); fi; end;; # # LatinSquareToOrthogonalArray(L) # # Input: A Latin square L # Output: The orthogonal array of L (the set of entries in L) # LatinSquareToOrthogonalArray:=function(L) local O,i,j; if(not IsLatinSquare(L)) then return fail; fi; O:=[]; for i in [1..Size(L)] do for j in [1..Size(L)] do Append(O,[[i,j,L[i][j]]]); od; od; return O; end;; # # PermutationCanonicalForm(alpha,n) # # Input: Permutation alpha of [1..n] # Output: Canonical permutation with the same cycle structure as alpha # PermutationCanonicalForm:=function(alpha,n) return CycleStructureToPermutation(Reversed(SortedList(List(Cycles(alpha,[1..n]),i->Size(i))))); end;; # # CanonicalConjugator(alpha,n) # # Input: Permutation alpha of [1..n] # Output: gamma such that gamma^(-1)*alpha*gamma=beta where # beta is the canonical form of alpha # # [Is there another name for this?] # CanonicalConjugator:=function(alpha,n) local beta,i,input,output,gamma; beta:=PermutationCanonicalForm(alpha,n); input:=Reversed(Permuted(Cycles(alpha,[1..n]),Sortex(List(Cycles(alpha,[1..n]),i->Size(i))))); output:=Cycles(beta,[1..n]); input:=Concatenation(input); output:=Concatenation(output); gamma:=List([1..n],i->i); for i in [1..n] do gamma[input[i]]:=output[i]; od; gamma:=PermList(gamma); if(gamma^(-1)*alpha*gamma=beta) then return gamma; fi; return fail; end;; # # LatinSquareWithCanonicalAutotopismPower(L,alpha,beta,gamma,p) # # Input: Latin square L that admits an autotopism (alpha,beta,gamma) and integer p # Output: returns a Latin square L' that admits the canonical autotopism # along with an autotopism (alpha',beta',gamma') of L' with the same # cycle structure as (alpha,beta,gamma)^p # LatinSquareWithCanonicalAutotopismPower:=function(L,alpha,beta,gamma,p) local a,b,c,n,M; if(not IsAutotopismLatinSquare(L,alpha,beta,gamma)) then return fail; fi; n:=Size(L); a:=CanonicalConjugator(alpha^p,n); b:=CanonicalConjugator(beta^p,n); c:=CanonicalConjugator(gamma^p,n); M:=IsotopicMatrix(L,a,b,c); a:=PermutationCanonicalForm(alpha^p,n); b:=PermutationCanonicalForm(beta^p,n); c:=PermutationCanonicalForm(gamma^p,n); if(IsLatinSquare(M) and IsAutotopismLatinSquare(M,a,b,c)) then return [M,a,b,c]; fi; return fail; end;; # # ThreeCyclesOffset1(d1,d2,d3,i) # ThreeCyclesOffset2(d1,d2,d3,i) # ThreeCyclesOffset3(d1,d2,d3,i) # # Offsets O_{i,j} described in the paper # ThreeCyclesOffset1:=function(d1,d2,d3,i) local g; if(d1 mod 2=0 and d1/2d) then return fail; fi; O:=[]; if(d mod 2=1) then for i in [1..d-f] do Append(O,[[i,d-i+1,1]]); od; for i in [1..f] do Append(O,[[d-f+i,f-i+1,d+i]]); od; for i in [1..f] do Append(O,[[d+i,f-i+1,1]]); Append(O,[[d-f+i,d+f-i+1,1]]); od; for i in [1..f] do for j in [1..f] do Append(O,[[d+i,d+j,d+1+((i+j) mod f)]]); od; od; fi; if(d mod 2=0) then if(d-f>=d/2) then for i in [1..d/2] do Append(O,[[i,d-i+1,1]]); od; for i in [1..d/2-f] do Append(O,[[d/2+i+1,d/2-i+1,1]]); od; for i in [1..f-1] do Append(O,[[d-f+i+1,f-i+1,d+i]]); od; for i in [1..f-1] do Append(O,[[d-f+i+1,d+f-i+1,1]]); od; Append(O,[[d/2+1,d+1,1]]); fi; if(d-fd) then return fail; fi; alpha:=CycleStructureToPermutation([d]); L:=ConstructionFromOrthogonalArray(GenerateOrthogonalArrayFromContourAndAutomorphism(OrthogonalArrayForContourWithCycleDAndFFixedPoints(d,f),alpha),d+f); if(IsLatinSquare(L) and IsAutomorphismLatinSquare(L,alpha)) then return L; fi; return fail; end;; # # Two non-trivial cycles # # # OrthogonalArrayProlongationTwoDistinctCycles(O,d1,d2,f,p) # # Input: This is a restricted type of prolongation used in these constructions # this function prolongates the contour for a Latin square with automorphism # with the cycle structure [d1,d2] or [d1,d2,1] to # one with the cycle structure [d1,d3,1,1,..,1] with f+p fixed points # (requires d1>d2) # OrthogonalArrayProlongationTwoDistinctCycles:=function(O,d1,d2,f,p) local i,j,k,O2; O2:=List(O,e->e); for k in [1..p] do i:=Filtered(O2,e->1<=e[1] and e[1]<=d1 and 1<=e[2] and e[2]<=d1 and 1<=e[3] and e[3]<=d1)[1]; Append(O2,[[d1+d2+f+k,i[2],i[3]]]); Append(O2,[[i[1],d1+d2+f+k,i[3]]]); Append(O2,[[i[1],i[2],d1+d2+f+k]]); O2:=Difference(O2,[i]); i:=Filtered(O2,e->1+d1<=e[1] and e[1]<=d1+d2 and 1+d1<=e[2] and e[2]<=d1+d2 and 1+d1<=e[3] and e[3]<=d1+d2)[1]; Append(O2,[[d1+d2+f+k,i[2],i[3]]]); Append(O2,[[i[1],d1+d2+f+k,i[3]]]); Append(O2,[[i[1],i[2],d1+d2+f+k]]); O2:=Difference(O2,[i]); od; if(f=1) then O2:=Difference(O2,[d1+d2+1,d1+d2+1,d1+d2+1]); fi; Append(O2,Filtered(OrthogonalArrayForContourWithCycleDAndFFixedPoints(d2,f+p),e->e[1]>d2 and e[2]>d2)+d1); return O2; end;; OrthogonalArrayForAutomorphismWithTwoCycles:=function(d1,d2,f) local O,O2,i,j,alpha,L; if(d10) then alpha:=CycleStructureToPermutation([2*d1]); L:=LatinSquareForAutomorphismWithCycleDAndFFixedPoints(2*d1,f); L:=LatinSquareWithCanonicalAutotopismPower(L,alpha,alpha,alpha,2)[1]; O:=LatinSquareToOrthogonalArray(L); return Filtered(O,e->e[3]=1 or e[3]=d1+1 or e[3]>2*d1); fi; if(d1=d2 and d1 mod 2=1) then L:=LatinSquareForAutomorphismWithCycleDAndFFixedPoints(d1,f); L:=DirectProductLatinSquares(L,[[1,2],[2,1]]); O:=LatinSquareToOrthogonalArray(L); return Filtered(O,e->e[3]=1 or e[3]=d1+1 or e[3]>2*d1); fi; O:=[]; # This contour is described in the ``equal cycle lengths'' theorem if(d1=d2 and d1 mod 2=0) then for i in [1..d1/2] do Append(O,[[d1/2-i+1,i,1],[3*d1/2-i+1,d1+i,1]]); od; for i in [1..d1/2-1] do Append(O,[[d1/2-i,i,d1+1],[3*d1/2-i,d1+i,d1+1]]); od; Append(O,[[d1,d1/2,d1+1],[2*d1,3*d1/2,d1+1]]); for i in [1..d1/2] do Append(O,[[d1-i+1,3*d1/2+i,1],[d1-i,3*d1/2+i,d1+1]]); Append(O,[[3*d1/2+i,d1-i+1,1],[3*d1/2+i-1,d1-i+1,d1+1]]); od; return O; fi; # In the following constructions we begin with f=0 or f=1 then prolong. if(d1>d2 and d1 mod 2=1) then # M(1,1) for i in [1..d1-d2] do Append(O,[[i,d1+1-i,1]]); od; for i in [1..d2] do Append(O,[[d1-d2+i,d2+1-i,d1+1]]); od; # M(1,2) for i in [1..d2] do Append(O,[[d1-d2+i,d1+d2+1-i,1]]); od; # M(2,1) for i in [1..d2] do Append(O,[[d1+i,d2+1-i,1]]); od; # subsquare M(2,2) O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d2,0); Append(O,O2+d1); return OrthogonalArrayProlongationTwoDistinctCycles(O,d1,d2,0,f); fi; if(d1>d2 and d1 mod 2=0 and d2 mod 2=1) then # M(1,1) Append(O,[[1,d1,1]]); for i in [1..d1/2-1] do Append(O,[[i+1,d1-i,1]]); od; for i in [1..d1/2-d2] do Append(O,[[d1/2+1+i,d1/2+1-i,1]]); od; for i in [1..d2-1] do Append(O,[[d1-d2+1+i,d2+1-i,d1+1]]); od; Append(O,[[1,1,d1+1]]); # M(1,2) Append(O,[[d1/2+1,d1+1,1]]); for i in [1..d2-1] do Append(O,[[d1-d2+1+i,d1+d2+1-i,1]]); od; # M(2,1) for i in [1..d2] do Append(O,[[d1+i,d2+1-i,1]]); od; # subsquare M(2,2) O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d2,0); Append(O,O2+d1); return OrthogonalArrayProlongationTwoDistinctCycles(O,d1,d2,0,f); fi; if(d1>d2 and d1 mod 2=0 and d2 mod 2=0) then # we require f>0; for now assume f=1 # M(1,1) for i in [1..d1/2-d2] do Append(O,[[i,d1-i+1,1]]); od; for i in [1..d2] do Append(O,[[d1/2-d2+i,d1/2+d2+1-i,d1+1]]); od; for i in [1..d1/2-1] do Append(O,[[d1/2+1+i,d1/2+1-i,1]]); od; Append(O,[[1,1,d1+d2+1]]); # M(1,2) for i in [1..d2/2] do Append(O,[[d1/2-d2+i,d1+d2+1-i,1]]); od; for i in [1..d2/2] do Append(O,[[d1/2-d2/2+1+i,d1+d2/2+1-i,1]]); od; # M(2,1) Append(O,[[d1+1,1,1]]); if(d1/d2 mod 2=1) then for i in [1..d2/2-1] do Append(O,[[d1+1+i,d1/2+d2/2+1-i,1]]); od; for i in [1..d2/2] do Append(O,[[d1+d2/2+i,d1/2+d2+1-i,1]]); od; else for i in [1..d2/2-1] do Append(O,[[d1+1+i,d1/2+d2+1-i,1]]); od; for i in [1..d2/2] do Append(O,[[d1+d2/2+i,d1/2+d2/2+1-i,1]]); od; fi; # M(1,infty) amd M(infty,1) Append(O,[[d1/2-d2/2+1,d1+d2+1,1]]); if(d1/d2 mod 2=1) then Append(O,[[d1+d2+1,d1/2+1,1]]); else Append(O,[[d1+d2+1,d1/2+d2/2+1,1]]); fi; # subsquare O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d2,1); Append(O,O2+d1); return OrthogonalArrayProlongationTwoDistinctCycles(O,d1,d2,1,f-1); fi; end;; ContourForAutomorphismWithTwoCycles:=function(d1,d2,f) if(d1d)); if(d mod 2=0 and m mod 2=1 and f=0) then return fail; elif(f>0) then beta:=CycleStructureToPermutation([d*m]); L:=LatinSquareForAutomorphismWithCycleDAndFFixedPoints(d*m,f); L:=LatinSquareWithCanonicalAutotopismPower(L,beta,beta,beta,m)[1]; elif(d mod 2=1 and f=0) then L:=LatinSquareForAutomorphismWithCycleDAndFFixedPoints(d,0); M:=List([1..m],i->List([1..m],j->(i+j) mod m))+1; L:=DirectProductLatinSquares(L,M); else # d mod 2=0 and f=0 L:=LatinSquareForAutomorphismWithTwoCycles(d,d,0); M:=List([1..m/2],i->List([1..m/2],j->(i+j) mod (m/2)))+1; L:=DirectProductLatinSquares(L,M); fi; if(IsLatinSquare(L) and IsAutotopismLatinSquare(L,alpha,alpha,alpha)) then return L; fi; return fail; end;; # # Three non-trivial cycles # # Cases i--iii of the d1>d2>d3 OrthogonalArrayForAutomorphismWithThreeCyclesPart1:=function(d1,d2,d3) local g,t1,t2,t3,infty,O,O2,i,s,n,j,t,f,r,c,h1,h2,e,X,alpha,L; # The case d1<=d2<=d3 with d2 even and d3 odd uses the construction with d2 and d3 swapped if(d2 mod 2=0 and d3 mod 2=1) then O:=OrthogonalArrayForAutomorphismWithThreeCyclesPart1(d1,d3,d2); alpha:=CycleStructureToPermutation([d1,d3,d2]); L:=ConstructionFromOrthogonalArray(GenerateOrthogonalArrayFromContourAndAutomorphism(O,alpha),d1+d2+d3+1); O:=LatinSquareToOrthogonalArray(LatinSquareWithCanonicalAutotopismPower(L,alpha,alpha,alpha,1)[1]); return Filtered(O,e->e[3]=1 or e[3]=d1+1 or e[3]=d1+d2+1 or e[3]>=d1+d2+d3+1); fi; g:=d2*d3/d1; t1:=1; t2:=d1+1; t3:=d1+d2+1; infty:=d1+d2+d3+1; n:=d1+d2+d3+1; if(d1 mod 2=0) then f:=1; else f:=0; fi; # M(1,1) s:=List([1..d1],i->1); for i in [1..d2-g] do s[i]:=t2; od; for i in [d2-g+1..d2+d3-2*g] do s[i]:=t3; od; if(d1 mod 2=0) then s[d1]:=infty; fi; O:=[]; for i in [1..d1] do Append(O,[[i,t2-i-ThreeCyclesOffset1(d1,d2,d3,i),s[i]]]); od; # M(1,2) and M(2,1) for i in [1..d2] do if(g0) then Append(O,[[Filtered([1..d1],i->not ForAny(O,o->o[1]=i and o[3]=1))[1],n,1]]); Append(O,[[n,Filtered([1..d1],i->not ForAny(O,o->o[2]=i and o[3]=1))[1],1]]); fi; # M(2,3) and M(3,2) if(d1 mod 2=1) then for i in [g+1..2*g] do Append(O,[[t3-3*g-1+i+ThreeCyclesOffset2(d1,d2,d3,i),n-i+ThreeCyclesOffset3(d1,d2,d3,i),t1]]); Append(O,[[t3-1+i-ThreeCyclesOffset3(d1,d2,d3,i),d1+3*g+1-i-ThreeCyclesOffset2(d1,d2,d3,i),t1]]); od; elif(d2 mod 2=1 and d3 mod 2=0) then # g is odd # M(2,3) h1:=0; h2:=d3/2-g; X:=Filtered([t2..t3],i->ForAll(O,e->e[1]<>i or e[3]<>1)); r:=X[1]; X:=Filtered([t3..n],i->ForAll(O,e->e[2]<>i or e[3]<>1)); c:=X[1]; Append(O,[[r,c,t1]]); for i in [1..g-1] do Append(O,[[r+g-i,c+h2+i,t1]]); od; # M(3,2) h1:=d3/2-g; h2:=0; X:=Filtered([t3..n],i->ForAll(O,e->e[1]<>i or e[3]<>1)); r:=X[Size(X)]-h1-1; X:=Filtered([t2..t3],i->ForAll(O,e->e[2]<>i or e[3]<>1)); c:=X[Size(X)]-h2-1; Append(O,[[r+h1+1,c+h2+1,t1]]); for i in [1..g-1] do Append(O,[[r-i+1,c-g+i+1,t1]]); od; elif(d2 mod 2=0 and d3 mod 2=0 and (d2-d3)/g mod 2=1) then # M(2,3) h1:=d2/2-g; h2:=d3/2-g; X:=Filtered([t2..t3],i->ForAll(O,e->e[1]<>i or e[3]<>1)); r:=X[1]; X:=Filtered([t3..n],i->ForAll(O,e->e[2]<>i or e[3]<>1)); c:=X[1]; Append(O,[[r,c,t1]]); for i in [1..g/2] do Append(O,[[r+h1+1+g/2+i-2,c+h2+1+g-i-1,t1]]); od; for i in [1..g/2-1] do Append(O,[[r+h1+1+i-1,c+h2+1+g/2-i-1,t1]]); od; # M(3,2) h1:=d3/2-g; h2:=d2/2-g; X:=Filtered([t3..n],i->ForAll(O,e->e[1]<>i or e[3]<>1)); r:=X[Size(X)]-h1-1; X:=Filtered([t2..t3],i->ForAll(O,e->e[2]<>i or e[3]<>1)); c:=X[Size(X)]-h2-1; Append(O,[[r+h1+1,c+h2+1,t1]]); for i in [1..g/2] do Append(O,[[r-g/2-i+2,c-g+i+1,t1]]); od; for i in [1..g/2-1] do Append(O,[[r-i+1,c-g/2+i+1,t1]]); od; fi; # subsquares (x2) O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d3,f); Append(O,O2+d1+d2); O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d2,f); # we shift the indices +d3 for the fixed rows, columns and symbols for t in O2 do t2:=[0,0,0]; for j in [1..3] do if(t[j]<=d2) then t2[j]:=t[j]; else t2[j]:=t[j]+d3; fi; od; Append(O,[t2+d1]); od; return O; end;; # Case v of the d1>d2>d3 OrthogonalArrayForAutomorphismWithThreeCyclesPart2:=function(d1,d2,d3) local g,t1,t2,t3,infty,O,i,s,n,O2,j,t,f,r,c,h1,h2,X; g:=d2*d3/d1; t1:=1; t2:=d1+1; t3:=d1+d2+1; infty:=d1+d2+d3+1; n:=d1+d2+d3+1; if(d1 mod 2=0) then f:=1; else f:=0; fi; # M(1,1) s:=List([1..d1],i->1); for i in [1..d2-g] do s[d1/2+1-i]:=d1+1; od; for i in [1..d3-g] do s[d1/2+i]:=d1+d2+1; od; if(d1 mod 2=0) then s[d1]:=infty; fi; O:=[]; for i in [1..d1] do Append(O,[[i,t2-i-ThreeCyclesOffset1(d1,d2,d3,i),s[i]]]); od; # M(1,2) for i in [1..d2] do if(gg) then Append(O,[[d2-2*g+i+d1/2-d2+g,n-i+ThreeCyclesOffset3(d1,d2,d3,i),t1]]); fi; fi; od; Append(O,[[d1,n-g+ThreeCyclesOffset3(d1,d2,d3,g)-d3/2,t1]]); # M(infty,1) and M(1,infty) if(f>0) then Append(O,[[Filtered([1..d1],i->not ForAny(O,o->o[1]=i and o[3]=1))[1],n,1]]); Append(O,[[n,Filtered([1..d1],i->not ForAny(O,o->o[2]=i and o[3]=1))[1],1]]); fi; # M(2,3) h1:=d2/2-g; h2:=0; X:=Filtered([t2..t3],i->ForAll(O,e->e[1]<>i or e[3]<>1)); r:=X[1]; X:=Filtered([t3..n],i->ForAll(O,e->e[2]<>i or e[3]<>1)); c:=X[1]; Append(O,[[r,c,t1]]); for i in [1..g/2] do Append(O,[[r+h1+1+g/2+i-2,c+h2+1+g-i-1,t1]]); od; for i in [1..g/2-1] do Append(O,[[r+h1+1+i-1,c+h2+1+g/2-i-1,t1]]); od; # M(3,2) h1:=0; h2:=d2/2-g; X:=Filtered([t3..n],i->ForAll(O,e->e[1]<>i or e[3]<>1)); r:=X[Size(X)]-h1-1; X:=Filtered([t2..t3],i->ForAll(O,e->e[2]<>i or e[3]<>1)); c:=X[Size(X)]-h2-1; Append(O,[[r+h1+1,c+h2+1,t1]]); for i in [1..g/2] do Append(O,[[r-g/2-i+2,c-g+i+1,t1]]); od; for i in [1..g/2-1] do Append(O,[[r-i+1,c-g/2+i+1,t1]]); od; # subsquares (x2) O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d3,f); Append(O,O2+d1+d2); O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d2,f); # we shift the indices +d3 for the fixed rows, columns and symbols for t in O2 do t2:=[0,0,0]; for j in [1..3] do if(t[j]<=d2) then t2[j]:=t[j]; else t2[j]:=t[j]+d3; fi; od; Append(O,[t2+d1]); od; return O; end;; # # OrthogonalArrayProlongationThreeDistinctCycles(O,d1,d2,d3,f,p) # # Input: This is a restricted type of prolongation used in these constructions # this function prolongates the contour for a Latin square with automorphism # with the cycle structure [d1,d2,d3] or [d1,d2,d3,1] to # one with the cycle structure [d1,d2,d3,1,1,..,1] with f+p fixed points # (requires d1>d2>d3) # OrthogonalArrayProlongationThreeDistinctCycles:=function(O,d1,d2,d3,f,p) local i,j,k,O2; O2:=List(O,e->e); for k in [1..p] do i:=Filtered(O2,e->1<=e[1] and e[1]<=d1 and 1<=e[2] and e[2]<=d1 and 1<=e[3] and e[3]<=d1)[1]; Append(O2,[[d1+d2+d3+f+k,i[2],i[3]]]); Append(O2,[[i[1],d1+d2+d3+f+k,i[3]]]); Append(O2,[[i[1],i[2],d1+d2+d3+f+k]]); O2:=Difference(O2,[i]); i:=Filtered(O2,e->1+d1<=e[1] and e[1]<=d1+d2 and 1+d1<=e[2] and e[2]<=d1+d2 and 1+d1<=e[3] and e[3]<=d1+d2)[1]; Append(O2,[[d1+d2+d3+f+k,i[2],i[3]]]); Append(O2,[[i[1],d1+d2+d3+f+k,i[3]]]); Append(O2,[[i[1],i[2],d1+d2+d3+f+k]]); O2:=Difference(O2,[i]); i:=Filtered(O2,e->1+d1+d2<=e[1] and e[1]<=d1+d2+d3 and 1+d1+d2<=e[2] and e[2]<=d1+d2+d3 and 1+d1+d2<=e[3] and e[3]<=d1+d2+d3)[1]; Append(O2,[[d1+d2+d3+f+k,i[2],i[3]]]); Append(O2,[[i[1],d1+d2+d3+f+k,i[3]]]); Append(O2,[[i[1],i[2],d1+d2+d3+f+k]]); O2:=Difference(O2,[i]); od; if(f=1) then O2:=Difference(O2,[[d1+d2+d3+1,d1+d2+d3+1,d1+d2+d3+1]]); fi; Append(O2,Filtered(OrthogonalArrayForContourWithCycleDAndFFixedPoints(d3,f+p),e->e[1]>d3 and e[2]>d3)+d1+d2); return O2; end;; # d1=d2>d3 and d3 even OrthogonalArrayForAutomorphismWithThreeCyclesPart3:=function(d1,d3,f) local O,i,O2,beta; if not (d1>d3 and d3 mod 2=0 and d1 mod d3=0 and f<=d3 and (f>0 or not (d3 mod 2=0 and d1/d3 mod 2=1))) then return fail; fi; O:=[]; # M(1,1) for i in [1..d1/2] do Append(O,[[i,d1-i,d1+1]]); od; for i in [1..d1/2-d3] do Append(O,[[i,d1-i+1,1]]); od; for i in [1..d3] do Append(O,[[d1/2-d3+i,d1/2+d3-i+1,2*d1+1]]); od; # M(1,2) for i in [1..f] do Append(O,[[d1/2+i,3*d1/2-i+1,2*d1+d3+i]]); od; for i in [1..d1/2-f] do Append(O,[[d1/2+f+i,3*d1/2-f-i+1,d1+1]]); od; for i in [1..d1/2] do Append(O,[[d1/2+i,3*d1/2-i+2,1]]); od; # M(2,1) for i in [1..f] do Append(O,[[3*d1/2+i,d1/2-i+1,2*d1+d3+i]]); od; for i in [1..d1/2-1] do Append(O,[[3*d1/2+i,d1/2-i,d1+1]]); od; for i in [1..d1/2-f] do Append(O,[[3*d1/2+f+i,d1/2-f-i+1,1]]); od; Append(O,[[2*d1,d1,d1+1]]); # M(2,2) Append(O,[[d1+1,d1+1,1]]); for i in [1..d1/2-d3] do Append(O,[[d1+i,2*d1-i+1,d1+1]]); od; for i in [1..d3] do Append(O,[[3*d1/2-d3+i,3*d1/2+d3-i+1,2*d1+1]]); od; for i in [1..d1/2-1] do Append(O,[[d1+i+1,2*d1-i+1,1]]); od; # M(1,3) for i in [1..d3/2] do Append(O,[[d1/2-d3+i,2*d1+d3-i+1,1]]); od; for i in [1..d3/2-1] do Append(O,[[d1/2-d3/2+i+1,2*d1+d3/2-i+1,1]]); od; Append(O,[[d1/2+1,2*d1+1,d1+1]]); # M(1,infty) Append(O,[[d1/2-d3/2+1,2*d1+d3+1,1]]); for i in [1..f-1] do Append(O,[[d1/2+1+i,2*d1+d3+f-i+1,d1+1]]); od; # M(2,3) for i in [1..d3/2] do Append(O,[[d1/2-d3+i+d1,2*d1+d3-i+1,d1+1]]); od; for i in [1..d3/2-1] do Append(O,[[d1/2-d3/2+i+1+d1,2*d1+d3/2-i+1,d1+1]]); od; Append(O,[[d1/2+1+d1,2*d1+1,1]]); # M(2,infty) Append(O,[[3*d1/2-d3/2+1,2*d1+d3+1,d1+1]]); for i in [1..f-1] do Append(O,[[3*d1/2+1+i,2*d1+d3+f-i+1,1]]); od; # M(3,1) for i in [1..d3/2] do Append(O,[[2*d1+i,d1/2+d3-i+1,1]]); od; for i in [1..d3/2] do Append(O,[[2*d1+d3/2+i,d1/2+d3/2-i,1]]); od; # M(infty,1) Append(O,[[2*d1+d3+1,d1/2+d3/2,1]]); for i in [1..f-1] do Append(O,[[2*d1+d3+i+1,d1/2-i,1]]); od; # M(3,2) for i in [1..d3/2] do Append(O,[[2*d1+i,3*d1/2+d3-i+1,d1+1]]); od; for i in [1..d3/2] do Append(O,[[2*d1+d3/2+i,3*d1/2+d3/2-i,d1+1]]); od; # M(infty,2) Append(O,[[2*d1+d3+1,3*d1/2+d3/2,d1+1]]); for i in [1..f-1] do Append(O,[[2*d1+d3+i+1,3*d1/2-i,d1+1]]); od; # add subsquare O2:=OrthogonalArrayForContourWithCycleDAndFFixedPoints(d3,f); Append(O,O2+2*d1); return O; end;; # d1>d2=d3 OrthogonalArrayForAutomorphismWithThreeCyclesPart4:=function(d1,d2,f) local O,i,j,L2,O2,s,x; if not(d1>=2*d2+f and d1 mod d2=0 and f<=2*d2 and (d2 mod 2=1 or d1/d2 mod 2=0 or f>0)) then return fail; fi; O:=[]; if(d1 mod 2=1) then # M(1,1) for i in [1..d1-2*d2-f] do Append(O,[[i,d1+1-i,1]]); od; for i in [1..d2] do Append(O,[[d1-2*d2-f+i,2*d2+f+1-i,d1+1]]); od; for i in [1..d2] do Append(O,[[d1-d2-f+i,d2+f+1-i,d1+d2+1]]); od; for i in [1..f] do Append(O,[[d1-f+i,f+1-i,d1+2*d2+i]]); od; # M(1,2), M(1,3) and M(1,infty) for i in [1..d2] do Append(O,[[d1-2*d2-f+2*i-1,d1+i,1]]); od; # M(1,2) for i in [1..d2] do Append(O,[[d1-2*d2-f+2*i,d1+d2+i,1]]); od; # M(1,3) for i in [1..f] do Append(O,[[d1-f+i,d1+2*d2+i,1]]); od; # M(1,infty) # M(2,1), M(3,1), M(infty,1) for i in [1..d2] do Append(O,[[d1+i,2*i-1,1]]); od; for i in [1..d2] do Append(O,[[d1+d2+i,2*i,1]]); od; for i in [1..f] do Append(O,[[d1+2*d2+i,2*d2+i,1]]); od; fi; if(d1 mod 2=0) then # M(1,1) # s describes the sequence used in the even pattern s:=List([1..d1],i->1); for i in [1..d2] do s[d1/2+1-i]:=d1+1; s[d1/2+i]:=d1+d2+1; od; if(f=0 and d1>2*d2) then s[d1]:=d1+d2+1; s[d1/2+d2]:=1; fi; if(f>0) then for i in [1..Minimum(f,d1/2-d2)] do s[d1+1-i]:=d1+2*d2+i; od; for i in [d1/2-d2+1..f] do s[d1+1-2*d2-i]:=d1+2*d2+i; od; fi; # Print(s,"\n"); for i in [1..d1/2] do Append(O,[[i,d1+1-i,s[i]]]); od; for i in [1..d1/2-1] do Append(O,[[d1/2+i,d1/2-i,s[d1/2+i]]]); od; Append(O,[[d1,d1,s[d1]]]); if(f=0) then # M(1,2) and M(1,3) for i in [1..d2] do Append(O,[[d1/2-d2+2*i-1,d1+i,1]]); od; # M(1,2) for i in [1..d2-1] do Append(O,[[d1/2-d2+2*i,d1+d2+i,1]]); od; # M(1,3) Append(O,[[d1,d1+2*d2,1]]); fi; if(f>0) then # M(1,2) and M(1,3) for i in [1..d2] do Append(O,[[d1/2-d2+2*i-1,d1+i,1]]); od; # M(1,2) for i in [1..d2] do Append(O,[[d1/2-d2+2*i,d1+d2+i,1]]); od; # M(1,3) # M(1,infty) for i in [1..Minimum(f,d1/2-d2)] do Append(O,[[d1+1-i,d1+2*d2+f+1-i,1]]); od; for i in [d1/2-d2+1..f] do Append(O,[[d1+1-2*d2-i,d1+2*d2+f+1-i,1]]); od; # M(infty,1) Append(O,[[d1+2*d2+1,d1/2-d2,1]]); x:=Minimum(f-1,d1/2-d2-1); for i in [1..x] do Append(O,[[d1+2*d2+1+i,i,1]]); od; for i in [1..f-x-1] do Append(O,[[d1+2*d2+1+x+i,d1/2+d2+i,1]]); od; fi; # M(2,1) and M(3,1) for i in [1..d2] do Append(O,[[d1+i,d1/2-d2+2*i-1,1]]); od; # M(2,1) for i in [1..d2] do Append(O,[[d1+d2+i,d1/2-d2+2*i,1]]); od; # M(3,1) fi; # add subsquare+ L2:=LatinSquareForAutomorphismWithTwoCycles(d2,d2,f); for i in [1..2*d2+f] do for j in [1..2*d2+f] do Append(O,[[i,j,L2[i][j]]+d1]); od; od; return O; end;; # # OrthogonalArrayForAutomorphismWithThreeCycles(d1,d2,d3,f) # ContourForForAutomorphismWithThreeCycles(d1,d2,d3,f) # LatinSquareForAutomorphismWithThreeCycles(d1,d2,d3,f) # # Input: d1, d2, d3 and f (with d1>=d2>=d3) # Output: An orthogonal array, contour or Latin square for automorphism # alpha with cycle structure [d1,d2,d3,1,1,...,1] with f fixed points # (assumes d1>=d2>=d3) # OrthogonalArrayForAutomorphismWithThreeCycles:=function(d1,d2,d3,f) local O,f2,alpha,L; if(d1e[3]=1 or e[3]=d1+1 or e[3]=2*d1+1 or e[3]>3*d1); elif(d1=d2 and d1>d3 and d3 mod 2=0) then return OrthogonalArrayForAutomorphismWithThreeCyclesPart3(d1,d3,f); elif(d1=d2 and d1>d3 and d3 mod 2=1) then alpha:=CycleStructureToPermutation([2*d1,d3]); L:=LatinSquareForAutomorphismWithTwoCycles(2*d2,d3,f); L:=LatinSquareWithCanonicalAutotopismPower(L,alpha,alpha,alpha,2)[1]; O:=LatinSquareToOrthogonalArray(L); return Filtered(O,e->e[3]=1 or e[3]=d1+1 or e[3]=2*d1+1 or e[3]>2*d1+d3); elif(d1>d2 and d2=d3) then return OrthogonalArrayForAutomorphismWithThreeCyclesPart4(d1,d2,f); elif(d1>d2 and d2>d3) then if(d1 mod 2=0) then f2:=1; else f2:=0; fi; if(d2 mod 2=0 and d3 mod 2=0 and (d2/Gcd(d2,d3)-d3/Gcd(d2,d3)) mod 2=0) then O:=OrthogonalArrayForAutomorphismWithThreeCyclesPart2(d1,d2,d3); else O:=OrthogonalArrayForAutomorphismWithThreeCyclesPart1(d1,d2,d3); fi; return OrthogonalArrayProlongationThreeDistinctCycles(O,d1,d2,d3,f2,f-f2); fi; return []; end;; ContourForForAutomorphismWithThreeCycles:=function(d1,d2,d3,f) if(d1theory_test) then Append(F,[[d1,f]]); Print([d1,f]," failed! Latin square found=",latin_test," Theory claims=",theory_test,"\n"); fi; for d2 in [2..d1] do alpha:=CycleStructureToPermutation([d1,d2]); L:=LatinSquareForAutomorphismWithTwoCycles(d1,d2,f); latin_test:=IsLatinSquare(L) and IsAutomorphismLatinSquare(L,alpha); theory_test:=ExistsLatinSquareTwoCycles(d1,d2,f); if(latin_test<>theory_test) then Append(F,[[d1,d2,f]]); Print([d1,d2,f]," failed! Latin square found=",latin_test," Theory claims=",theory_test,"\n"); fi; for d3 in [2..d2] do alpha:=CycleStructureToPermutation([d1,d2,d3]); L:=LatinSquareForAutomorphismWithThreeCycles(d1,d2,d3,f); latin_test:=IsLatinSquare(L) and IsAutomorphismLatinSquare(L,alpha); theory_test:=ExistsLatinSquareThreeCycles(d1,d2,d3,f); if(latin_test<>theory_test) then Append(F,[[d1,d2,d3,f]]); Print([d1,d2,d3,f]," failed! Latin square found=",latin_test," Theory claims=",theory_test,"\n"); fi; od; od; od; od; return F; end;; LatinSquaresContourTest2:=function(n) local d1,d2,d3,f,L,alpha,latin_test,theory_test,F; F:=[]; for d1 in [2..n] do Print(d1,"\n"); for f in [0..1] do for d2 in [2..d1] do for d3 in [2..d2] do alpha:=CycleStructureToPermutation([d1,d2,d3]); L:=LatinSquareForAutomorphismWithThreeCycles(d1,d2,d3,f); latin_test:=IsLatinSquare(L) and IsAutomorphismLatinSquare(L,alpha); theory_test:=ExistsLatinSquareThreeCycles(d1,d2,d3,f); if(latin_test<>theory_test) then Append(F,[[d1,d2,d3,f]]); Print([d1,d2,d3,f]," failed! Latin square found=",latin_test," Theory claims=",theory_test,"\n"); fi; od; od; od; od; return F; end;;