Element structure of direct product of D8 and Z2

We use the following presentation, where $$e$$ denotes the identity element:

$$G := \langle a,x,y \mid a^4 = x^2 = y^2 = e, xax = a^{-1}, ya = ay, xy = yx \rangle$$.

Conjugacy class structure
The equivalence classes up to automorphisms are: