Let $G$ be the group given by the set of invertible matrices of the form \begin{bmatrix}a & b & c\\0 & d & e\\0 & 0 & f\end{bmatrix} with $a,b,c,d,e,f \in \mathbb Z_3$.
Find the composition length of $G$ and its composition factors in terms of known groups, specifying which groups occur as composition factors and how many times each occurs in the composition series.
Attempt: I also know that the subset $N$ of $G$ of matrices where $a=d=f=1$ along the diagonal is a normal subgroup of $G$, that the centre of $N$, $Z(N)$ is \begin{bmatrix}1 & 0 & c\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} and that $G$ is soluble. Can I use this information to answer the question?