Enumeration of chiral and achiral edge and face substitutions of buckminsterfullerene

The edge and face groups of the buckminsterfullerene cluster are formulated. The generalized character cycle indices of all the irreducible representations of both edge and face groups are obtained. The generating functions computed from these groups yield the chiral and achiral isomers, and mappings that transform according to the specified irreducible representation for edge and face colorings of buckminsterfullerene.