Enumeration of left braces with additive group \(C_2 \times C_2 \times C_4 \times C_4\) (Q6635329)

Left braces are algebraic structures that include radical rings that were introduced by \textit{W. Rump} [J. Algebra 307, No. 1, 153--170 (2007; Zbl 1115.16022)] to study involutive non-degenerate solutions of the Yang-Baxter equation. Also, any abelian group is a (trivial) brace. Classifying and describing all left braces of a fixed order is a difficult task as it is for groups.\N\NThe determination of the isomorphism classes of left braces with additive group isomorphic to a given abelian group \(G\) reduces to the determination of all conjugacy classes of regular subgroups of the holomorph of \(G\). With this, the numbers of isomorphism classes of left braces of all orders up to \(120\) except \(32, 64, 81\), and \(96\) are presented in [\textit{L. Guarnieri} and \textit{L. Vendramin}, Math. Comput. 86, No. 307, 2519--2534 (2017; Zbl 1371.16037)].\N\NIn this paper, the authors provide a complete and extraordinary classification of left braces of order \(64\). In particular, they show there are as many \(10,326,821\) isomorphism classes of left braces with additive group isomorphic to \(C_2 \times C_2 \times C_4 \times C_4\).