Distributive and trimedial quasigroups of order 243

Abstract

We enumerate three classes of non-medial quasigroups of order 243 = 3(5) up to isomorphism. There are 17 004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Beneteau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Nemec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Oprsal and Stanovsky).

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 LOOPS package in GAP.

Keywords

• Distributive quasigroup
• Trimedial quasigroup
• Medial quasigroup
• Entropic quasigroup
• Commutative Moufang loop
• Latin square
• Mendelsohn triple system
• Classification
• Enumeration