Representing the Quotient Groups of a Finite Permutation Group

Let G be a permutation group of finite degree d. We prove that the product of the orders of the composition factors of G that are not alternating groups acting naturally, in a sense that will be made precise, is bounded by cd − 1/d, where c = 4.5. We use this to prove that any quotient G/N of G has a faithful permutation representation of degree at most cd − 1.