Computing matrix representations

Let $G$ be a finite group and $\chi$ a faithful irreducible character for $G$. Earlier papers by the first author describe techniques for computing a matrix representation for $G$ which affords $\chi$ whenever the degree $\chi (1)$ is less than $32$. In the present paper we introduce a new, fast method which can be applied in the important case where $G$ is perfect and the socle $soc(G/Z(G))$ of $G$ over its centre is abelian. In particular, this enables us to extend the general construction of representations to all cases where $\chi (1)\leq 100$. The improved algorithms have been implemented in the new version 3.0.1 of the GAP package REPSN by the first author.