This manual describes the **Repsn** package for computing matrix representations in characteristic zero of finite groups. Most of the functions in **Repsn** have been written according to the algorithm described in the author's Ph.D thesis [DA03] and [DD10] (see [DA05]).

For constructing representations of simple groups and their covers we use the algorithm described in [Dix93]. To use this algorithm for constructing a representation of a group \(G\) affording an irreducible character \(\chi\) of \(G\), we need to have a subgroup \(H\) of \(G\) such that the restriction of \(\chi\) to \(H\) has a linear constituent with multiplicity one. In this case we say \(H\) is a *character subgroup* relative to \(\chi\) (or a \(\chi\)-subgroup). A \(\chi\)-subgroup for each irreducible character \(\chi\) of degree less than 100 of simple groups and their covers are listed in [DA06] and [DA07].

All **Repsn** functions are written entirely in the **GAP** language. It is proved in [DA05] and [DD10] that the algorithm is correct for any group with a character of degree less than 100. Indeed, if the group is solvable, there is no restriction on the character degree. In practice the program is quite fast when the degree is small, but can be very slow when it is necessary to call one of the subprograms which extend irreducible representations. In the latter case the number of element wise operations required to extend a representation of degree \(d\) is proportional to \(d^6\).

**Repsn** is implemented in the **GAP** language, and runs on any system supporting **GAP**4. The **Repsn** package is loaded into the current **GAP** session with the command

gap> LoadPackage( "repsn" );

(see section *Loading a GAP Package* in the **GAP** Reference Manual).

Please report any bugs or other issues you might encounter via the **Repsn** issue tracker at https://github.com/gap-packages/repsn/issues.

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