# Composition Series

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?

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Hint: By a direct computation you can show that $Z(N)$ is also the commutator subgroup of $N$. In particular, $N/Z$ is abelian, so you are reduced to studying $G/N$. To study this, look for a map $G \to D$, where $D$ is the group of diagonal matrices. (If "composition series" means the quotients occurring must be cyclic, and not just abelian, you will also need to study $D$ and $N/Z$ a little more closely.)