Let H = {u, v,w, x, y, z} be a group (with multiplication *) and $\theta$ : $S_3$ - H be an isomorphism with $\theta$(e) = u, $\theta$(1, 2) = x, $\theta$(1, 3) = y, $\theta$(2, 3) = z,$\theta$(1, 2, 3) = v ,$\theta$(1, 3, 2) = w. Find x * w, $w^{−1}$, $v^5$, z * v$^{−1}$ * x.
I know for x * w, I have to calculate it by (1,2)(1,3,2).
However I'm not sure how to go any further than this.
For x*w, as you said, calculate $(1,2)(1,3,2)=(1,3)$ and then apply $\theta$ on it to get $\theta{(1,3)}=y$ , now $w^{-1}=(1,3,2)^{-1}=(2,3,1)$. Now you can do rest two yourself.