For sufficiently large n consider this question Let $G$ be a finite group with the following properties:

$|G|=n!$

$H,K$ are subgroups of $G$ such that $H\cap K=1$ and $H\cong S_{3}$ and $K\cong S_{n-3}$

- $t$ is an involution in $H$ such that $|C_G(t)|=2(n-2)!$

Now, is it true that $C_G(t)\cong \langle t \rangle \times S_{n-2}$