Centraliser of a fixed-point-free involution
= Centraliser of a fixed-point-free involution
{title2=$C_{S_{2m}}((1\ 2)\cdots(2m-1\ 2m))\cong C_2\wr_{\{1,\ldots,m\}}S_m$}
Commutation with a fixed-point-free involution preserves its partner relation. A commuting <permutation> can permute the $m$ pairs and independently swap each pair. This gives the <permutation wreath product> with order $2^m m!$.