D8 in SD16

Definition
Here, $$G$$ is the semidihedral group:SD16, the semidihedral group of order sixteen (and hence, degree eight). We use here the presentation:

$$G := \langle a,x \mid a^8 = x^2 = e, xax = a^3 \rangle$$

$$G$$ 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 $$\langle a^2,x \rangle$$. It is dihedral of order 8 and is given by:

$$\! \{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}$$

The multiplication table of $$H$$ is as follows:

Cosets
The subgroup has index two and is hence a normal subgroup (See index two implies normal). Its left cosets coincide with its right cosets. There are two cosets:

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

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

G := SmallGroup(16,8); H := Filtered(Subgroups(G), x -> Order(x) = 8 and IsDihedralGroup(x))[1];

The GAP display is as follows:

gap> G := SmallGroup(16,8); H := Filtered(Subgroups(G), x -> Order(x) = 8 and IsDihedralGroup(x))[1];  Group([ f4, f3, f2 ])

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

gap> Order(G); 16 gap> Order(H); 8 gap> Index(G,H); 2 gap> StructureDescription(H); "D8" gap> StructureDescription(G/H); "C2" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true