### Abstract

Various forms, including multi-dimensional forms of the cycle
index of the symmetry group of the fullerene C_{60}
acting on its sets of vertices, faces, edges etc. are computed and
applied for the enumeration of *essentially different*
colourings of the truncated icosahedron. Furthermore we determine
the 3-dimensional cycle index of a polyhedron under the
*leapfrog operation* introduced by P.W. Fowler.