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
$$<a,b,c : a^{p^{2}}=b^p=c^p=e, ab=ba^{1+p},ac=cab,bc=cb>$$
From the above relations, I can produce following relation
$$a^ib^j=b^ja^{{(1+p)}^ji},$$ Now I am trying to obtain similar relation for the generators $a$ and $c$. (Here $i$ and $j$ are natural numbers). Kindly help me out as I am not able to proceed further to write $a^ic^j= ? ?$ in simple manner.



Presentations of groups of order $p^4$
$$a^ic^j = c^j(ab^j)^i.$$
I expect you could use your existing formula for $a^ib^j$ to write $(ab^j)^i$ in the form $b^ka^l$, but I will leave that to you!


