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I need to type the group Figure 1

in GAP. Is it correct if I type as, "SemidirectProduct(ZmodnZ(3),AbelianGroup([7,7]));"? Will it recognize that AbelianGroup([7,7]) is the normal subgroup?


marked as duplicate by Matthew Towers, Ivan Neretin, Jeremy Rickard, hardmath, ArsenBerk Oct 4 '18 at 19:03

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  • $\begingroup$ Here I have considered the internal Semidirect product $\endgroup$ – Buddhini Angelika Sep 27 '18 at 10:16
  • $\begingroup$ The suggested duplicate can give you a good start on the GAP syntax for this. Note that an internal semidirect product involves a subgroup $H$ "acting" on a normal subgroup $N$ with trivial intersection. It doesn't appear to me that the syntax you've shown allows for this action to be determined, even in principle. $\endgroup$ – hardmath Oct 4 '18 at 15:20

You will need to also specify the action of $Z_3$ on $Z_7^2$. Generically, you would do this by specifying as second argument (i.e. arguments are subgroup, map, normal subgroup) a homomorphism from $Z_3$ into the automorphism group os $Z_7^2$.

As your normal subgroup is a vector space, this whole construction however is easier done by describing the action through matrices. for example (not sure whether this is the action you want):

gap> m:=[[4,0],[0,2]]*One(GF(7));
[ [ Z(7)^4, 0*Z(7) ], [ 0*Z(7), Z(7)^2 ] ]
gap> s:=SemidirectProduct(Group(m),GF(7)^2);
<matrix group of size 147 with 3 generators>

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