Center of semidihedral group:SD16

The semidihedral group $$SD_{16}$$ (also denoted $$QD_{16}$$) is the semidihedral group (also called quasidihedral group) of order $$16$$. Specifically, it has the following presentation:

$$G = SD_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle$$.

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

The quotient group is isomorphic to dihedral group:D8.

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.

Note that the multiplication table for the quotient group looks identical to that for center of dihedral group:D16, although the original groups differ (semidihedral group:SD16 versus dihedral group:D16).