CentralExtensionXMod( f )
This construction returns the crossed module whose boundary f is a surjection from S to R having as kernel a subgroup of the centre of S. The action maps an element of r in R to conjugation of S by f^{-1}r.
gap> d8 := Subgroup( s4, [ (1,2,3,4), (1,3) ] );; d8.name := "d8";; gap> gend8 := d8.generators;; genk4 := k4.generators;; gap> f := GroupHomomorphismByImages( d8, k4, gend8, genk4 );; gap> EX := CentralExtensionXMod( f ); Crossed module [d8->v4] gap> XModPrint( EX ); Crossed module [d8->v4] :- : Source group d8 has parent s4 and generators: [ (1,2,3,4), (1,3) ] : Range group k4 has parent s4 and generators: [ (1,2)(3,4), (1,3)(2,4) ] : Boundary homomorphism maps source generators to: [ (1,2)(3,4), (1,3)(2,4) ] : Action homomorphism maps range generators to automorphisms: (1,2)(3,4) --> { source gens --> [ (1,2,3,4), (2,4) ] } (1,3)(2,4) --> { source gens --> [ (1,4,3,2), (1,3) ] } These 2 automorphisms generate the group of automorphisms.
GAP 3.4.4