In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, *"Theory of groups of finite order"*. The group ($\mathbb{Z}\_{p^{2}}\rtimes \mathbb{Z}\_{p^{}}) \rtimes\_{\phi}\mathbb{Z}\_{p^{}} $, have presentation
$$\langle a,b,c : a^{p^{2}}=b^p=c^p=e,\,ab=ba^{1+p},\,ac=cab,\,bc=cb\rangle$$


I was trying to explore about the representation (must be faithful) of the above group in $\mathrm{GL}(4,F\_p)$, i.e. group of $4\times 4$ invertible matrices, taken over $F\_p$. Please guide me in the right direction, from where to start or if my statement mathematically correct or not ?



Faithful representation of group of order $p^4$
One small remark: if $p > 4$ then any element $x$ of ${\rm GL}(4,F\_{p})$ of order a power of $p$ satisfies $(x-I)^{4} = 0$, so that we certainly have $(x-I)^{p} = 0$ and $x^{p} = I$.
Hence ${\rm GL}(4,F\_{p})$ contains no element of order $p^{2}$ when the prime $p$ is greater than $3$.


