Distributive and trimedial quasigroups of order 243
  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}. 
P\v{r}emysl Jedli\v{c}ka, David Stanovsk\'y, Petr Vojt\v{e}chovsk\'y
math.GR