Semisymmetric Latin Squares

This data was produced in my collaboration with Brendan McKay for the paper "Enumeration of Latin squares with conjugate symmetry".

A semisymmetric Latin square is one that is equal to 3 of its conjugates (also known as parastrophes). That means that the set of (row,column,symbol) triples is unchanged by cyclic permutation of each triple.

We start with catalogues of semisymmetric Latin squares up to isomorphism:

...and the species that contain semisymmetric Latin squares:

Next, isomorphism classes of idempotent semisymmetric Latin squares:

order 3
order 4
order 7
order 9
order 10
(There are none of orders 2, 5, 6, 8 or 11).

Next, isomorphism classes of diagonal semisymmetric Latin squares:

order 3
order 4
order 5
order 7
order 8
order 9
order 10
order 11
(There are none of orders 2 or 6).

Next, isomorphism classes of semisymmetric loops (reduced LS):

order 2
order 4
order 5
order 8
order 10
order 11
(There are none of orders 3, 6, 7 or 9).

We did also generate the semisymmetric Latin squares of order 11, the idempotent semisymmetric Latin squares of order 12, the diagonal semisymmetric Latin squares of order 12 and the semisymmetric loops of order 12. In each case, there are too many to present in this format, but they are available on request.

If you want to know what format these files are in, it is my usual latin squares format.

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