Suppose we have a group isomorphism $G \cong H \leq S_{q+1} $.
Let $ \chi_{q+1} $ denote the number of $(q+1)$ - cycles $\in H$ and suppose $q$ is an odd prime, then I've shown that $ \frac{q-1}{2} \leq \chi_{q+1} \leq q! $. Without any further information, can I draw any conclusions on what types of cycles is contained in $H$ and how many there is of each?
Probably a bit vague, is it solvable if $ \chi_{q+1} $ is known?
For instance, assume $ | H | = q(q^2-1)$, and the number of $(q+1)$-cycles is $x$. What other elements does $H$ contain?