| | | **The symmetry group of the fullerene C**_{60} |

## The symmetry group of the fullerene C_{60}

The icosahedron and the C_{60}-fullerene

From the way of constructing the truncated icosahedron
from the *icosahedron*
(see see figure)
it is clear that they both have the
same symmetry group *S* which is
isomorphic to *A*_{5}´S_{2}.
The *rotational symmetries* form a
subgroup *R* of *S* of
index 2 which is isomorphic to *A*_{5}.
The point group symbols for *S* and *R* are
I_{h} and I.
The elements of *R* can be described in the following way:
The 12 pentagonal
faces of the fullerene C_{60} form 6 pairs of opposite faces.
A rotation axis through
the centres of two such faces is an axis of a 5-fold rotation.
The 20 hexagonal faces of the fullerene C_{60} form 10 pairs
of opposite faces through the
centres of which we each have an axis of a 3-fold rotation.
Finally the set of
edges can be partitioned into 60 *pentagonal
edges* surrounding the
pentagonal faces and 30 *hexagonal edges*
lying between two hexagons.
The hexagonal edges form 15 pairs of opposite edges the centres of which
each determine an axis of a 2-fold rotation.
The full symmetry group of the fullerene C_{60}
is computed by combining these rotations
with one reflection of the fullerene.

In the present paper the symmetries of a fullerene are described as
permutations of its set of vertices.
This permutation representation induces
a group action on the sets of all
edges or faces by identifying an edge or a
face with the vertices incident to it.
The generators of *R* will be
indicated by *p*_{1} and *p*_{2},
whereas *s* stands for a reflection.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

| | | **The symmetry group of the fullerene C**_{60} |