Element structure of dihedral group:D16

This article describes the structure of elements of the dihedral group:D16, which is given by the following presentation:

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

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

Elements
Below, we list all the elements, also giving the interpretation of each element under the geometric description of the dihedral group as the symmetries of a 8-gon, and for the corresponding permutation representation (see D16 in S8). Note that for different conventions, one can obtain somewhat different correspondences, so this may not match up with other correspondences elsewhere. Note that the descriptions below assume the left action convention for functions and the corresponding convention for composition, and hence some of the entries may become different if you adopt the right action convention.: