Given a number field and a rational function , one may take pre-images of , maybe in or transcendental over , under successive iterates of , and thus obtain an infinite -ary tree rooted at by assigning an edge between two pre-images , if . The absolute Galois group of K acts on by tree automorphisms and gives rise to a subgroup of the group of all automorphisms of . We find a new class of natural overgroups in which the image of this Galois representation attached to a PCF polynomial must live. Following this result, we find that the image of the Galois representation of a new PCF polynomial is isomorphic to one of the overgroups. We also study the structure of the overgroups for specific maps, including normalized dynamical Belyi maps, and exhibit that the normal subgroups of the overgroups form a unique chief series. Consequently, we can bound the number of generators in terms of group-theoretic analysis.