Derived subgroup of dihedral group:D16

Definition
Here, $$G$$ is the dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation:

$$G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle$$

$$G$$ has 16 elements:

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

The subgroup $$H$$ of interest is the subgroup $$\langle a^2 \rangle$$. It is cyclic of order 4 and is given by:

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

The quotient group is a Klein four-group.

Cosets
The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets:

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

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

G := DihedralGroup(16); H := DerivedSubgroup(G);

The GAP display looks as follows:

gap> G := DihedralGroup(16); H := DerivedSubgroup(G);  Group([ f3, f4 ])

Here is a GAP implementation to verify some of the assertions made in this page:

gap> StructureDescription(G/H); "C2 x C2" gap> Order(G); 16 gap> Order(H); 4 gap> Index(G,H); 4 gap> IsNormal(G,H); true gap> H = FrattiniSubgroup(G); true gap> H = Agemo(G,2,1); true gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true