Given a graph order of $4n$ for positive $n$, find graphs with $4^n (2n)!$ and $2(2n-1)!^2$ automorphisms. I already have graphs for the first, but haven't had any luck on the other one.
I figured that it suffices to find a graph of order $2m$ with $2(m-1)!^2$ automorphisms, because $4n=2\cdot 2m$. I got the first one by guessing some well known graphs, but that doesn't seem to be working too well for the second.
I'm not necessarily looking for a solution. More so a hint or possible ideas for graphs to look at.