Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there ezists a function $J(d)$ of a positive integer $d$ for which the following holds: if $G$ is a finite group having a faithful linear representation over $\mathbb C$ of degree $d$, then $G$ has a normal Abelian subgroup $A$ with $[G:A]\le J(d)$. We show that if $G$ is a transitive permutation group and $d$ is the maximal degree of irreducible representations of $G$ entering its permutation representation, then there exists a normal solvable subgroup $A$ of $G$ such that $[G:A]\le J(d)^{\log_2d}$. Bibliography: 7 titles.