Permutation Groups · Superchallenge problem

$\sigma \in S_9$ has cycle type $(5,4)$: $\sigma=(1\,2\,3\,4\,5)(6\,7\,8\,9)$, so $|\sigma|=20$. How many of the $20$ elements of the cyclic subgroup $\langle\sigma\rangle$ are ODD permutations?