Center of dihedral group:D16

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 $$\langle a^4 \rangle = \{ e, a^4 \}$$.

The quotient group is isomorphic to dihedral group:D8.

The subgroup is isomorphic to cyclic group:Z2 and the multiplication table is given below. Note that the group is an abelian group, so we do not need to worry about left/right issues:

Cosets
The subgroup has order 2 and index 8, so it has 8 left cosets. It is a normal subgroup, so the left cosets coincide with the right cosets. The cosets are:

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

The quotient group is isomorphic to dihedral group:D8, and the multiplication table on cosets is given below. The row element is multiplied on the left and the column element is multiplied on the right.

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

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

The GAP display looks as follows:

gap> G := DihedralGroup(16); H := Center(G);  Group([ f4 ])

Here is a GAP implementation to verify some of the assertions made on this page: gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> StructureDescription(G/H); "D8" gap> H = CommutatorSubgroup(G,DerivedSubgroup(G)); true gap> H = Socle(G); true gap> H = Agemo(G,2,2); true gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true