Let $G$ be a transitive permutation group on a set of size $n$, and suppose $Z(G)=1$ (for instance $G$ is a direct power of a nonabelian simple group). What can we say about the centraliser $K$ of $G$ in $Sym(n)$? I'm interested firstly if there are any restrictions on $K$ independent of degree, and secondly on what role the degree plays.

Of course, there is the classical result that $C_{Sym(n)}(G)$ is a semiregular subgroup of $Sym(n)$ of cardinality $Fix(G_{0})$, where $G_{0}$ is the stabilizer of a point and $Fix(G_{0})$ is the set of points fixed by $G_{0}$. 

