The action on the sets of edges and faces Some cycle indices The symmetry group of the fullerene C60 The action on the set of vertices

The action on the set of vertices

Labelling the vertices of the fullerene C60, given by its Schlegel diagram

Labelling the vertices of the fullerene C60 as is indicated in figure the permutation representation of generators of R and S acting on the set of vertices is given by:

π1= (56,57,58,59,60)(51,52,53,54,55)(42,44,46,48,50)(41,43,45,47,49)(32,34,36,38,40) (31,33,35,37,39)(22,24,26,28,30)(21,23,25,27,29)(12,14,16,18,20)(11,13,15,17,19) (6,7,8,9,10)(1,2,3,4,5)

π2= (37,38,48,54,47)(27,28,49,59,46)(18,39,60,53,26)(17,29,55,58,36)(10,40,56,45,16) (9,19,50,57,35)(7,12,22,23,13)(5,30,51,44,15)(4,20,41,52,25)(3,6,31,43,24) (2,11,32,33,14)(1,21,42,34,8)

σ= (58)(57,59)(56,60)(53)(52,54)(51,55)(45,46)(44,47)(43,48)(42,49)(41,50)(35,36) (34,37)(33,38)(32,39)(31,40)(25,26)(24,27)(23,28)(22,29)(21,30)(15,16)(14,17)(13,18) (12,19)(11,20)(8,9)(7,10)(6)(3,4)(2,5)(1)

π1 is a 5-fold rotation around the centre of figure and π2 is a 5-fold rotation, where the rotation axis goes through the centres of the faces {7,12,22,23,13} and {37,38,48,54,47}.

There are computer algebra systems (e.g. SYMMETRICA [18]) which evaluate the cycle index of a permutation group given by a set of generators. In this way the cycle indices of R and S acting on the set of vertices of C60 are computed as

Z(R,Vertices)=(1)/(60)(24 x512 +20 x320 +15 x230 +x160) ..
Z(S,Vertices)=(1)/(120)(24 x106 +20 x610 +24 x512 +20 x320 +16 x230 +15 x14 x228 +x160). ..
harald.fripertinger@kfunigraz.ac.at

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