Another possibility to get a pc group in GAP is to convert a polycyclic group given by some other representation to a pc group. For finitely presented groups there are various quotient methods available. For all other types of groups we use the following procedures.
PcGroupWithPcgs( mpcgs ) A
creates a new Pc group G whose family pcgs is isomorphic to the (modulo) pcgs mpcgs.
returns a pc group with family pcgs corresponding to pcgs.
gap> G := Group( (1,2,3), (3,4,1) );; gap> PcGroupWithPcgs( Pcgs(G) ); <pc group of size 12 with 3 generators> gap> G := SymmetricGroup( 5 ); Sym( [ 1 .. 5 ] ) gap> H := Subgroup( G, [(1,2,3,4,5), (3,4,5)] ); Group([ (1,2,3,4,5), (3,4,5) ]) gap> modu := ModuloPcgs( G, H ); [ (4,5) ] gap> PcGroupWithPcgs(modu); <pc group of size 2 with 1 generators>
IsomorphismPcGroup( G ) A
returns an isomorphism from G onto an isomorphic PC group. The series chosen for this PC representation depends on the method chosen. G may be a polycyclic group of any kind, for example a permuattion group.
gap> G := Group( (1,2,3), (3,4,1) );; gap> iso := IsomorphismPcGroup( G ); Pcgs([ (2,4,3), (1,2)(3,4), (1,3)(2,4) ]) -> [ f1, f2, f3 ] gap> H := Image( iso ); <pc group of size 12 with 3 generators>
IsomorphismSpecialPcGroup( G ) A
returns an isomorphism from G onto an isomorphic PC group whose family pcgs is a special pcgs. (This can be beneficial to the runtime of calculations.) G may be a polycyclic group of any kind, for example a permuattion group.
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GAP 4 manual