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doi:10.1016/j.jsc.2005.01.002 
Copyright © 2005 Elsevier Ltd All rights reserved.
An algorithm for constructing representations of finite groups
Received 16 November 2003;
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Abstract
Let G be a finite group. It is easy to compute the character of G corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical representation.
Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, we know of no general method which has been proposed which is practical for any but small groups.
We shall describe an algorithm for computing an irreducible matrix representation which affords a given character χ of a given group G. The algorithm uses properties of the structure of G which can be computed efficiently by a program such as GAP, theoretical results from representation theory, theorems from group theory (including the classification of finite simple groups), and linear algebra. All results in this paper have been implemented in the GAP package REPSN.
Keywords: Algorithm; Perfect groups; Irreducible representations
MSC: primary; 20C40; secondary; 20C15
