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

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.