The polytopes of the H group with 60 vertices and their orbit decompositions.

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter-Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H is given and presented as a `pancake structure'.