D8 in S4

This article is about the subgroup $$H$$ in the group $$G$$, where $$G$$ is symmetric group:S4, i.e., the symmetric group on the set $$\{ 1,2,3,4 \}$$, and $$H$$ is the subgroup:

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

$$H$$ is a 2-Sylow subgroup of $$G$$ and is isomorphic to dihedral group:D8. It has two other conjugate subgroups, which are given below:

$$\! H_1 = \{, (1,3,2,4), (1,2)(3,4), (1,4,2,3), (1,3)(2,4), (1,4)(2,3), (1,2), (3,4) \}$$

and:

$$\! H_2 = \{, (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}$$

Complements
The permutable complements to $$H$$ (and also to each of its conjugates) are precisely the subgroups of order three, namely A3 in S4 and its conjugates. Each of these is a conjugate to each of the conjugates of $$H$$:

$$\! \{, (1,2,3), (1,3,2) \}, \{ (1,3,4), (1,4,3) \}, \{ , (1,2,4), (1,4,2) \}, \{ , (2,3,4), (2,4,3) \}$$

In addition, any S2 in S4 is a lattice complement but not a permutable complement to two of the three conjugates of $$H$$, namely the ones that do not contain it.

Effect of subgroup operators
In the table below, we provide values specific to $$H$$.

Fusion system on subgroup
The subgroup embedding induces the non-inner non-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 symmetric group:S4. 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 SymmetricGroup and SylowSubgroup functions as follows:

G := SymmetricGroup(4); H := SylowSubgroup(G,2);

Note that this doesn't output the subgroup $$H$$ used on this page, but rather the conjugate subgroup $$H_1$$. However all relevant properties are invariant under conjugation, so this does not matter.