conjugacy action of a subgroup of a finite group on the lattice of subgroup in GAP

Question

Let $G$ be a finite group with the lattice of subgroups ${\mathcal L}(G)$ and let $H$ be a subgroup of $G$, which acts on ${\mathcal L}(G)$ by conjugation. That is for each $h\in H$ and each $T\in {\mathcal L}(G)$, define $T.h:=h^{-1}Th$ . How could we determine this action in a GAP code and compute the orbits of this action? (For example let $G:=S_4$, the symmetric group on four letters
and $H:= A_4$, the alternating group on four letters).

Any answer will be greatly appreciated!

Answer 1

You first get the subgroups of $G$ in a list, for example from the conjugacy classes of subgroups

gap> G:=SymmetricGroup(4);;H:=AlternatingGroup(4);;
gap> subs:=Concatenation(List(ConjugacyClassesSubgroups(G),AsList));;

The action by conjugation is by the ^ operator, so

gap> Orbits(H,subs);
[ [ Group(()) ],
  [ Group([ (3,4) ]), Group([ (1,4) ]), Group([ (2,4) ]), Group([ (1,2) ]),
  Group([ (2,3) ]), Group([ (1,3) ]) ], [...]

gives you the orbits. Similarly Action (or ActionHomomorphism) gives you the permutation action image.