Element structure of nontrivial semidirect product of Z4 and Z4

This article gives information on the element structure of the group:

$$G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$$

where $$e$$ denotes the identity element.

Conjugacy class structure
The equivalence classes up to automorphisms are:

Directed power graph
Below is a collapsed version of the directed power graph of the group. Each node represets an equivalence class of elements that all generate the same cyclic subgroup. There is an edge from one vertex to another if the latter is the square of the former. A dashed edge means that the latter is an odd power of the former. We remove all the loops.