The structure of a group of order 443520

Question

Let $G$ be a finit group and $T$ be a normal subgroup of $G$ such that $PSL(3,4) \unlhd T \leq Aut(PSL(3,4))$ and $|T|=2|PSL(3,4)|$. If $G= T\rtimes C_{11}$, then what we can say about the structure of the group $G$? (Actually I would like $G$ has an element of order 22 and by $C_n$ I mean the cyclic group of order $n$).

Answer 1

$G=T\times C$ using the same ideas as before. How does $C$ act on the socle $S$ of $T$? not by an element of order 11, but rather it centralizes it. It centralizes $T/S$ as well. Heck it centralizes $G/T$, so $C$ centralizes a chief series of $G$, and so is contained in the Fitting subgroup of $G$.