In 1985 the first C60 clusters were produced and it was assumed
that its 60 carbon atoms are placed at the vertices of a truncated
icosahedron (which is commonly called a foot ball or soccer ball.)
The truncated icosahedron is one of the 14 Archimedean solids
already known to the ancient Greeks.
Five years later 
the original hypotheses on the structure of C60 could be confirmed.
A nice introduction into the theory of the fullerenes
providing historical and chemical background information
can be found in .
From the mathematical point of view in the present paper
we are interested in the
combinatorics of the symmetry group of the truncated icosahedron and
we will discuss some examples in how many ways the soccer ball
can be coloured in essentially different ways.
From the chemical point of view we will determine the number of all
possible placements of double bonds for the C60 fullerene.
Furthermore investigating the C60H60-molecule
(i.e. a hypothetic molecule where all the thirty double bonds
of the C60 are hydrated),
we will compute the numbers of all
Finally the implementation of these cycle index methods
in the computer algebra system SYMMETRICA 
will be discussed and some further cycle indices will be listed.
last changed: January 23, 2001