2 improve expression

Sorry, I should elaborate a bit here. I am working on a graph $\Gamma$ which admits a metacyclic group transitive on its vertices. By some other restricted conditions, I have reduced that $Aut\Gamma$ is contained in $AGL(4,3)$. So it turns out that I need to work on the following problems:

• Check whether there exist such metacyclic subgroups in $AGL(4,3)$.
• If exist, what are they?
• One of my attempt was to inspect subgroups of $AGL(4,3)$ and rule out those impossible cases. But it seems tedious and somehow I still have to seek help from machinery search...

Alternatively, I am hoping to do it on a more elegant and efficient way. That is, to find the generators of the metacyclic groups. We know that a metacyclic group can be generated by two elements. Hence if we can find all the possible generators, then we can construct the groups and verify if these groups are transitive or not. There are some helpful information we already know: written $AGL(4,3)=\mathbb{Z}_3^4{:}GL(4,3)$, so a transitive metacyclic group $G=\langle(a_1,b_1),(a_2,b_2)|a_i\in\mathbb{Z}_3^4,b_i\in GL(4,3),~\mbox{and}~o(b_1),o(b_2)~\mbox{divisible by}~9\rangle$. Then the next question is which exact generators we should choose?

Another thought is that if it is possible to do a pure computer search? as the group is not too big, but I am not too sure how to write up the codes...

Any suggestion would be much appreciated.

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# Metacyclic groups in $AGL(4,3)$

Just wonder if there is any known results on the transitive metacyclic groups of $AGL(4,3)$?