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
$$<a,b,c : a^{p^{2}}=b^p=c^p=e, ab=ba^{1+p},ac=cab,bc=cb>$$
I was trying to find the embedding of the above group into a symmetric group of order $p^4$, which exists by Cayley's Theorem. Is it possible to explicitly find the map ? Kindly see it.



Permutation representation of a finite $p$-group
Using the relations, you can represent every element uniquely as $c^k b^l a^m$ with $0\leq k < p$, $0\leq l < p$, $0\leq m< p^2$. Now you can work out how left multiplication with $a, b, c$ acts on the set of such representatives by again using the relations, this is easy to do explicitly. The corresponding action on the set of tuples $(k, l, m) $ as above is the desired embedding.


