Let H be the subgroup of $\displaystyle GL(3, \mathbb{Z}_3)$ consisting of all matrices of the form $\displaystyle \left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right]$, where $\displaystyle a,b,c \in \mathbb{Z}_3$. I have to prove that Z(H) is isomorphic to $\displaystyle \mathbb{Z}_3$ and that $\displaystyle H/Z(H)$ is isomorphic to $\displaystyle \mathbb{Z}_3 \times \mathbb{Z}_3$.

I'm really not sure how to begin with this. I started by taking two arbitrary matrices h and k from H and doing hk = kh to see what a matrix in Z(H) would have to look like, but I didn't really get anywhere with that. My initial instinct would be to just define a mapping from Z(H) to $\displaystyle \mathbb{Z}_3$, but I'm not sure how to do that, since I can't figure out what's in Z(H). Is there a better way to do this?