Derived subgroup of M16

Definition
We consider the group:

$$G = M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$$

with $$e$$ denoting the identity element.

This is a group of order 16, with elements:

$$\{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$$

We are interested in the subgroup:

$$H = \{ e, a^4 \}$$

This is a cyclic subgroup of order two, i.e., it is isomorphic to cyclic group:Z2. The quotient group is isomorphic to the abelian group direct product of Z4 and Z2.

Cosets
The subgroup is a normal subgroup and hence its left cosets coincide with its right cosets. The eight cosets are given below:

$$\{ e, a^4 \}, \{ a, a^5 \}, \{ a^2, a^6 \}, \{ a^3, a^7 \}, \{x, a^4x \}, \{ ax, a^5x \}, \{ a^2x, a^6x \}, \{ a^3x, a^7x \}$$

Complements
The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.

GAP implementation
The group and subgroup can be constructed using GAP's SmallGroup and DerivedSubgroup functions as follows:

 G := SmallGroup(16,6); H := DerivedSubgroup(G);

The GAP display looks as follows:

gap> G := SmallGroup(16,6); H := DerivedSubgroup(G);  Group([ f4 ])

Here is some GAP implementation to verify assertions made on this page:

gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true