Element structure of semidihedral group:SD16

This article describes the structure of elements of the semidihedral group:SD16, which is given by the following presentation:

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

Here, $$e$$ denotes the identity element. Every element is of the form $$a^k, 0 \le k \le 7$$ or $$a^kx, 0 \le k \le 7$$.

Conjugacy class structure
The equivalence classes up to automorphisms are:

Order and power information
The graph below is a collapsed and trimmed version of the directed power graph of the group. Here, we collapse together all elements that generate the same cyclic subgroup, and an edge from one vertex to another is drawn if the latter is the square ofthe former. We omit the loop at the identity element.