# 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 62.33.24 on the rows and columns, and 63.31.21.16 on the symbols.
• A latin square of order 41 that has an autotopism with cycle structure 64.33.24 on the rows and columns, and 65.31.21.16 on the symbols.
• A latin square of order 131 that has an autotopism with cycle structure 301.152.103.66.51 on the rows 302.103.61.51.310 on the columns, and 302.152.61.51.215 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 23801.17851.14281.10201.4201.11 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 90091.64351.50051.40951.34651 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