   The fullerene C20

## The fullerene C20

The smallest fullerene is C20, which has no hexagonal faces. The atoms of the C20 are positioned at the vertices of a pentagon dodecahedron, so the symmetry group of C20 is isomorphic to the symmetry group of C60. The generators for R(C20), the group of all rotational symmetries, acting on the set of vertices are given by
p1=(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)
and
p2=(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(C20) 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
Z3(R(C20))=(1)/(60)( v120e130f112 + 20v12v36e310f34 + 15v210e12e214f26 + 24v54e56f12f52)
Z3(S(C20))=(1)/(2) Z3(R(C20)) + (1)/(120) ( v210e215f26 + 20v2v63e65f62 + 15v14v28e14e213f14f24 + 24v102e103f2f10).

harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001   The fullerene C20