D8 in A6

This article is about the subgroup $$H$$ in the group $$G$$, where $$G$$ is alternating group:A6, i.e., the alternating group on the set $$\{ 1,2,3,4,5,6 \}$$, and $$H$$ is the subgroup:

$$\! H = \{, (1,2,3,4)(5,6), (1,3)(2,4), (1,4,3,2)(5,6), (1,2)(3,4), (1,4)(2,3), (1,3)(5,6), (2,4)(5,6) \}$$

$$H$$ is a 2-satisfies property::Sylow subgroup of $$G$$ and is isomorphic to dihedral group:D8. There is a total of 45 conjugate subgroups to $$H$$ in $$G$$ (including $$H$$ itself).

Fusion system
The subgroup embedding induces the simple fusion system for dihedral group:D8.

Construction of subgroup given group as a black box
Suppose we are already given a group $$G$$ that we know to be isomorphic to alternating group:A6. Then, the subgroup $$H$$ can be constructed using SylowSubgroup as follows:

H := SylowSubgroup(G,2);

Construction of group-subgroup pair
The group and subgroup pair can be defined using GAP's AlternatingGroup and SylowSubgroup functions as follows:

G := AlternatingGroup(6); H := SylowSubgroup(G,2);