Universal Covers of Finite Groups

Motivated by the success of quotient algorithms, such as the well-known p-quotient or solvable quotient algorithms, in computing information about finite groups, we describe how to compute finite extensions H ˜ of a finite group H by a direct sum of isomorphic simple Z p H -modules such that H and H ˜ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of H. Defining this covering group requires a study of the representation module, as introduced by Gaschutz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. An important application of our results is that they can be used to compute, for a given epimorphism G → H and simple Z p H -module V, the largest quotient of G that maps onto H with kernel isomorphic to a direct sum of copies of V. For this we also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group H, assuming a confluent rewriting system for H. To represent the corresponding group extensions on a computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.