Kurt Ewald wrote:
theory says that SmallGroup(12,1) is the semidirect product of z3 by z4
Group([ (1,2,3) ])
Group([ (1,2,3,4) ])
<group of size 2 with 1 generators>
Group([ (2,3)(4,5,6,7), (1,2,3) ])
<pc group of size 12 with 3 generators>
[ (2,3)(4,5,6,7), (1,3,2) ] -> [ f1, f3 ]
The complement (z4) of z3 must be a subgroup of the semiproduct.
Where lies the error?
This behaviour arises from the treatment of subgroups by GAP -
so GAP considers also, for example, the factors of a
direct product not as subgroups :
gap> z3 := CyclicGroup(3);; z4 := CyclicGroup(4);; gap> G := DirectProduct(z3,z4);; gap> IsSubgroup(G,z3); false gap> IsSubgroup(G,z4); false
Hope this helps,