Non-central Z4 in 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:

$$H := \{ e, a^2x, a^4, a^6x \}$$

This subgroup is isomorphic to cyclic group:Z4, with generator $$a^2x$$. It is a normal subgroup and the quotient group is also isomorphic to the cyclic group:Z4.

Here is the multiplication table for $$H$$. Note that $$H$$ is an abelian group so we don't have to worry about left/right issues:

Cosets
The subgroup is a normal subgroup and so its left cosets are the same as its right cosets. It has 4 cosets in the whole group:

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

Below is the multiplication for $$G/H$$. It is a cyclic group of order 4 with generator the coset $$\{ a, a^3x, a^5, a^7x \}$$. Since it is an abelian group, we do not need to worry about left/right convention for the multiplication table:

GAP implementation
The group and subgroup pair can be constructed as follows:

G := SmallGroup(16,6); H := Filtered(NormalSubgroups(G),x -> Order(x) = 4 and IsCyclic(x) and not(x = Center(G)))[1];

Here is the GAP display:

gap> G := SmallGroup(16,6); H := Filtered(NormalSubgroups(G),x -> Order(x) = 4 and IsCyclic(x) and not(x = Center(G)))[1];  Group([ f2*f3, f4 ])

Here is GAP code to verify some of the assertions in this page:

gap> Order(G); 16 gap> Order(H); 4 gap> Index(G,H); 4 gap> StructureDescription(H); "C4" gap> StructureDescription(G/H); "C4" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false