Centraliser of a transposition
= Centraliser of a transposition
{title2=$C_{S_n}((1\ 2))\cong C_2\times S_{n-2}$}
A <permutation> commutes with a transposition exactly when it preserves its two-element support as a set. It may swap those two points and independently permute the remaining points. Thus this <centraliser> has order $2(n-2)!$.