In theory pc groups are finitely presented groups. In practice the arithmetic in pc groups is different from the arithmetic in fp groups. Thus for technical reasons the pc groups in GAP do not form a subcategory of the fp groups and hence the methods for fp groups cannot be applied to pc groups in general.
IsPcGroup( G ) C
tests whether G is a pc group.
gap> G := SmallGroup( 24, 12 ); <pc group of size 24 with 4 generators> gap> IsPcGroup( G ); true gap> IsFpGroup( G ); false
Note that it is possible to convert a pc group to a fp group in GAP. The following function computes the power-commutator presentation defined by pcgs. The string str can be used to give a name to the generators of the fp group.
IsomorphismFpGroupByPcgs( pcgs, str )
gap> p := FamilyPcgs( SmallGroup( 24, 12 ) ); [ f1, f2, f3, f4 ] gap> iso := IsomorphismFpGroupByPcgs( p, "g" ); [ f1, f2, f3, f4 ] -> [ g1, g2, g3, g4 ] gap> F := Image( iso ); <fp group on the generators [ g1, g2, g3, g4 ]> gap> RelatorsOfFpGroup( F ); [ g1^2, g2^-1*g1^-1*g2*g1*g2^-1, g3^-1*g1^-1*g3*g1*g4^-1*g3^-1, g4^-1*g1^-1*g4*g1*g4^-1*g3^-1, g2^3, g3^-1*g2^-1*g3*g2*g4^-1*g3^-1, g4^-1*g2^-1*g4*g2*g3^-1, g3^2, g4^-1*g3^-1*g4*g3, g4^2 ]
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GAP 4 manual