Computing 3-Dimensional Groups : Crossed Squares and Cat$^2$-Groups

TL;DR

This paper explores the computational aspects of 3-dimensional algebraic structures called crossed squares and cat$^2$-groups, providing algorithms, implementation details, and extensive classification data.

Contribution

It introduces algorithms for computing with crossed squares and cat$^2$-groups, and provides a classification of 1,000 isomorphism classes up to order 30.

Findings

Algorithms for computing with crossed squares and cat$^2$-groups are detailed.

A comprehensive table of 1,000 isomorphism classes up to order 30.

Implementation in the GAP package $ extsf{XMod}$ is described.

Abstract

The category $\mathbf{XSq}$ of crossed squares is equivalent to the category $\mathbf{Cat2}$ of cat$^2$-groups. Functions for computing with these structures have been developed in the package $\mathsf{XMod}$ written using the $\mathsf{GAP}$ computational discrete algebra programming language. This paper includes details of the algorithms used. It contains tables listing the $1,000$ isomorphism classes of cat$^2$-groups on groups of order at most $30$.