Does $|x^G|$ divide the order $\langle x^G\rangle$? Let $G$ be a finite group, $x\in G$, $x^G$ denotes the conjugacy class contained $x$, that's to say
$|x^G|=n$. My question is:
what is the relationship between $|x^G|$ and the subgroup $\langle x^G\rangle$? what's more, I want to know whether $n$ divides the order of the subgroup $\langle x^G\rangle$?

Take $G=S_3$, $x=(123)$, then $x^G=\{(123),(213)\}$, whereas $\langle x^G\rangle=\{id,(123),(213)\}$. So the answer to the divisibility question is no.
