Here is the problem that I have.
Let $C=\{a=(ijkl)\}$ be the set of all cycles of length 4 in the symmetric group $S_4$. $S_4$ acts on the set $C$ by conjugation. For every cycle $a\in C$ determine the stabilizer in $S_4$. Are they all conjugated?
I think I am missing something here, as $C$ is of length 4, so there is no way that $gag^{-1}=a$ where $a\in C$ and $g\in S_4$.
EDIT: As one of the guys in comments section said, $a\in stab_{S_4}(a)$ and as stabilizer is a group, we have $a^2, a^3 \in stab_{S_4}(a)$. Until now, I have $stab_{S_4}(a)\in \{e, a, a^2, a^3\}$. Is there any other permutation in stab? I don't want to check all 24 permutations, as it is not interesting.