  We enumerate three classes of non-medial quasigroups of order $243=3^5$ up to
isomorphism. There are $17004$ non-medial trimedial quasigroups of order $243$
(extending the work of Kepka, B\'en\'eteau and Lacaze), $92$ non-medial
distributive quasigroups of order $243$ (extending the work of Kepka and
N\v{e}mec), and $6$ non-medial distributive Mendelsohn quasigroups of order
$243$ (extending the work of Donovan, Griggs, McCourt, Opr\v{s}al and
Stanovsk\'y).
  The enumeration technique is based on affine representations over commutative
Moufang loops, on properties of automorphism groups of commutative Moufang
loops, and on computer calculations with the \texttt{LOOPS} package in
\texttt{GAP}.

Group Theory