Cycle structure of autotopisms of quasigroups and Latin squares

An autotopism of a Latin square is a triple $(\alpha ,\beta ,\gamma )$ of permutations such that the Latin square is mapped to itself by permuting its rows by $\alpha$, columns by $\beta$, and symbols by $\gamma$. Let $\mathrm{A}\mathrm{t}\mathrm{p}(n)$ be the set of all autotopisms of Latin squares of order $n$. Whether a triple $(\alpha ,\beta ,\gamma )$ of permutations belongs to $\mathrm{A}\mathrm{t}\mathrm{p}(n)$ depends only on the cycle structures of $\alpha$, $\beta$ and $\gamma$. We establish a number of necessary conditions for $(\alpha ,\beta ,\gamma )$ to be in $\mathrm{A}\mathrm{t}\mathrm{p}(n)$, and use them to determine $\mathrm{A}\mathrm{t}\mathrm{p}(n)$ for $n\le 17$. For general $n$ we determine if $(\alpha ,\alpha ,\alpha )\in \mathrm{A}\mathrm{t}\mathrm{p}(n)$ (that is, if $\alpha$ is an automorphism of some quasigroup of order $n$), provided that either $\alpha$ has at most three cycles other than fixed points or that the non-fixed points of $\alpha$ are in cycles of the same length.