I have $G$ a finite group with a subgroup $H$ of index 7 and $C$ a cyclic group of order 2. I want to define in GAP the semidirect product $C^7 \rtimes G$ where every $g \in G$ acts on $C^7$ by permuting the coordinates in the same way $g$ permutes the cosets of $G/H$. How can I define this group? In particular I don't manage to find the elements of $Aut(C^7)$ corresponding to the action of $S_7$ that permutes the coordinates.
What you want to construct is called a Wreath Product. You can construct it in GAP by specifying the group $C$ and the permutation action of $G$. For example:
gap> C:=CyclicGroup(2); <pc group of size 2 with 1 generators> gap> G:=SmallGroup(21,1); # some group with subgroup of index 7 <pc group of size 21 with 2 generators> gap> H:=SylowSubgroup(G,3); # has index 7 Group([ f1 ]) gap> hom:=FactorCosetAction(G,H); <action epimorphism> gap> Image(hom); # indeed 7 points Group([ (2,3,5)(4,7,6), (1,2,3,4,5,6,7) ]) gap> W:=WreathProduct(C,G,hom); <group of size 2688 with 3 generators>
If you wanted to use the semi direct product construction you would need to construct an automorphism of $C^7$ that maps the components cyclically.