From notes, I think all of the following are true:
- Every element of $D_6$ can be written as $s^ir^j$, where $i = 0,1$ and $0\le j\le 5$.
- $r^6 = e$, where $e$ is the identity.
- $s^2 = e$
- $r^ks = sr^{-k}$ for any integer $k$.
Do I actually have to find $z$? If not, how would I prove its existence? I'm guessing I would have to suppose I have some $z$ such that $zg = gz$ and then have to show it is an element of $D_6$?