ENUMERATION OF CAT¹-GROUPS OF LOW ORDER

Abstract

In this paper we describe a share package [Formula: see text] of functions for computing with finite, permutation crossed modules, cat 1 -groups and their morphisms, written using the [Formula: see text] group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat 1 -groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module [Formula: see text] , and the corresponding monoid of sections of a cat 1 -group [Formula: see text] , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of [Formula: see text] , are used to construct the actor crossed module of [Formula: see text] which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat 1 -structures on groups of order at most 30.