Enumeration of left braces with additive group $C_2\times C_2\times C_4\times C_4$

TL;DR

This paper completes the classification of left braces of order 64 with a specific additive group structure, identifying over ten million isomorphism classes, which advances understanding of algebraic structures related to solutions of the Yang-Baxter equation.

Contribution

It provides the exact count of isomorphism classes of left braces of order 64 with a given additive group, completing the classification for this order.

Findings

Number of isomorphism classes of such braces is 10,326,821.

Total isomorphism classes of braces of order 64 is 15,095,601.

The classification fills a gap in the understanding of algebraic structures related to the Yang-Baxter equation.

Abstract

We show that the number of isomorphism classes of left braces of order~$64$ with additive group isomorphic to $C_2\times C_2\times C_4\times C_4$ is $10\,326\,821$. This completes the classification of left braces of order~$64$, that turn out to fall into $15\,095\,601$ isomorphism classes.