Let $\sigma=(1,2,3,\dots,n)$ be an odd $n-$cycle in $S_n$ (so $n$ is even). It is known that the size of its conjugacy class is $|cl_{S_n}(\sigma)|=(n-1)!$. I am interested in the size of the subset $S=cl_{cl_{S_n}(\sigma)}(\sigma)$, that is, the set of all the $n-$cycles that we can obtain by co...