## The fullerene C_{20}

The smallest fullerene is C_{20}, which has no hexagonal faces.
The atoms of the C_{20} are positioned at the vertices
of a *pentagon dodecahedron*, so the symmetry group
of C_{20} is isomorphic to the symmetry group of C_{60}.
The generators for *R(C*_{20}), the group of all rotational
symmetries, acting on the set of vertices are given by
*p*_{1}=(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)

and
*p*_{2}=(1,7,17,19,10)(2,12,18,14,5)(3,8,13,9,4)(6,11,16,20,15).

To get the group *S(C*_{20})
we have to add a third generator, a reflection
*s=(1)(2,5)(3,4)(6)(7,10)(8,9)(11,15)(12,14)(13)(16,20)(17,19)(18).*

The 3-dimensional cycle indices for the action on the sets of vertices,
edges and faces are
*Z*_{3}(R(C_{20}))=**(**1**)/(**60**)**(
v_{1}^{20}e_{1}^{30}f_{1}^{12} +
20v_{1}^{2}v_{3}^{6}e_{3}^{10}f_{3}^{4} +
15v_{2}^{10}e_{1}^{2}e_{2}^{14}f_{2}^{6} +
24v_{5}^{4}e_{5}^{6}f_{1}^{2}f_{5}^{2})

*
Z*_{3}(S(C_{20}))=**(**1**)/(**2**)** Z_{3}(R(C_{20})) +
**(**1**)/(**120**)**
( v_{2}^{10}e_{2}^{15}f_{2}^{6} +
20v_{2}v_{6}^{3}e_{6}^{5}f_{6}^{2} +
15v_{1}^{4}v_{2}^{8}e_{1}^{4}e_{2}^{13}f_{1}^{4}f_{2}^{4} +
24v_{10}^{2}e_{10}^{3}f_{2}f_{10}).

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001