Computing matrix representations. Let G be a finite group and χ a faithful irreducible character for G. Earlier papers by the first author describe techniques for computing a matrix representation for G which affords χ whenever the degree χ(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 χ(1)≤100. The improved algorithms have been implemented in the new version 3.0.1 of the GAP package REPSN by the first author.