The Moufang loops of order 64 and 81

TL;DR

This paper classifies Moufang loops of orders 64 and 81, revealing a large number of nonassociative structures beyond the known groups, using a linear algebraic approach to central loop extensions.

Contribution

It provides a complete classification of Moufang loops of orders 64 and 81, including the enumeration of nonassociative cases, using a novel linear algebraic method.

Findings

4262 nonassociative Moufang loops of order 64

5 nonassociative Moufang loops of order 81

2 commutative Moufang loops of order 81

Abstract

We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the $15$ groups of order $81$, there are $5$ nonassociative Moufang loops of order $81$, $2$ of which are commutative.