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Dear GAP Forum,

Katsushi Waki wrote

I want to know how to translate characters from quotinet group.

I mean that let G is a finite group and N is a normal subgroup of G.

I want to use characters of G/N as characters of G.

This is described in the section "Restricted and Induced Class Functions"

in the GAP Reference Manual,

which states that inflating characters of a factor group can be done

using `RestrictedClassFunction' or `RestrictedClassFunctions'.

The connection between the group $G$ and its factor group $F$ must be made

explicit either by a homomorphism from $G$ to $F$ or by the fact that

the factor fusion is stored on the character table of $G$.

Here is an example.

gap> g:= SymmetricGroup( 4 ); Sym( [ 1 .. 4 ] ) gap> n:= DerivedSubgroup( g );; gap> hom:= NaturalHomomorphismByNormalSubgroup( g, n ); [ (1,2,3,4), (1,2) ] -> [ f1, f1 ] gap> f:= Image( hom ); Group([ f1 ]) gap> firr:= Irr( f ); [ Character( CharacterTable( Group([ f1 ]) ), [ 1, 1 ] ), Character( CharacterTable( Group([ f1 ]) ), [ 1, -1 ] ) ] gap> RestrictedClassFunctions( firr, hom ); [ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1 ] ) ]

And here is the variant with the stored factor fusion.

First we compute and store it ...

gap> fus:= FusionConjugacyClasses( hom ); [ 1, 2, 1, 1, 2 ] gap> gtbl:= CharacterTable( g ); CharacterTable( Sym( [ 1 .. 4 ] ) ) gap> ftbl:= CharacterTable( f ); CharacterTable( Group([ f1 ]) ) gap> StoreFusion( gtbl, fus, ftbl );

... and now we can inflate characters also as follows.

gap> RestrictedClassFunctions( firr, gtbl ); [ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1 ] ) ]

I hope this helps.

Kind regards,

Thomas

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