Prove that the order of the centralizer $C(κ) ⊂ S_n$ is $6(n − 5)!$

Question

Let $κ = (123)(45) ∈ S_n$, where n ≥ 5. Prove that the order of the centralizer $C(κ) ⊂ S_n$ is $6(n − 5)!$ I think I am meant to use the Orbit-Stabilizer Theorem here but I am not sure how to apply it.

Answer 1

If $\pi$ commutes with $\kappa$ then for all $x$ occurring in a $k$-cycle of $\kappa$, $\pi(x)$ also occurs in a $k$-cycle of $\kappa$. Consequently, $\pi$ is some permutation of $\{1,2,3\}$, together with some permutation of $\{4,5\}$, together with some permutation of $\{6,\ldots,n\}$. We readily verify that we can take an arbitrary permutation of $\{6,\ldots,n\}$ (i.e., the centralizer of the identity in $S_{n-5}$ is all of $S_{n-5}$), and an arbitrary permutation of $\{4,5\}$ (i.e., the centralizer of the non-trivial element of $S_2$ is all of $S_2$), but only an even permutation of $\{1,2,3\}$ (i.e., the centralizer of $(1\,2\,3)\in S_3$ is $A_3$). We conclude that $C(\kappa)$ is isomorphic to $A_3\times S_2\times S_{n-5}$.