Latin squares with specified autotopisms
From this page you can download
Latin squares of orders up to
17 that have any specified autotopism (at least, you can specify
the cycle structure of the autotopism, from this it is easy to get any
autotopism you might want). These squares were constructed by my
former PhD student Doug Stones using gap code which can be downloaded here. That code can be
used to create larger examples with specified autotopisms.
This work was performed for our paper:
D. S. Stones, P. Vojtechovský and I. M. Wanless,
Cycle structure of autotopisms of quasigroups and Latin squares.
There are also some specific larger examples displaying interesting behaviour.
 A latin square of order 29
that has an autotopism with cycle structure
6^{2}.3^{3}.2^{4} on the rows and columns,
and 6^{3}.3^{1}.2^{1}.1^{6} on the symbols.
 A latin square of order 41
that has an autotopism with cycle structure
6^{4}.3^{3}.2^{4} on the rows and columns,
and 6^{5}.3^{1}.2^{1}.1^{6} on the symbols.
 A latin square of order 131
that has an autotopism with cycle structure
30^{1}.15^{2}.10^{3}.6^{6}.5^{1}
on the rows
30^{2}.10^{3}.6^{1}.5^{1}.3^{10}
on the columns, and
30^{2}.15^{2}.6^{1}.5^{1}.2^{15}
on the symbols.
These three examples answer a question about autotopisms of
Latin squares of prime order, asked in the paper of Stones et al.
[above]. Another question from that paper is answered by the following
examples of quasigroups possessing an automorphism of order more than
the order of the quasigroup.

A latin square of order 7034
that has an automorphism with cycle structure
2380^{1}.1785^{1}.1428^{1}.1020^{1}.420^{1}.1^{1}
Thus the order of the automorphism is 7140. Note that since the example
is quite big, only the first rows from each of the 5 long row cycles is given.
The other rows can easily be recovered from these, using the automorphism,
and knowing that the last row and column are in order.

A latin square of order 28009
that has an automorphism with cycle structure
9009^{1}.6435^{1}.5005^{1}.4095^{1}.3465^{1}
Thus the order of the automorphism is 45005. Again, since the example
is so big, only the first rows from each of the 5 row cycles is given.
The other rows can easily be recovered from these, using the automorphism.
While you are here you might also be interested in
the equivalent page
for autoparatopisms
Latin squares with transitive
autotopism groups
Back to Latin squares data homepage.