Universal covers of finite groups

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