Fullerenes -- The fullerene C20

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

π₁=(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)^._.

and

π₂=(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

σ=(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 "at" uni-graz.at, May 10, 2016

The fullerene C₂₀