| | | **The action on the set of vertices** |

## The action on the set of vertices

Labelling the vertices of the fullerene C_{60},
given by its Schlegel diagram

Labelling the vertices of the fullerene C_{60}
as is indicated in
figure the
permutation representation of generators
of *R* and *S* acting on
the set of vertices is given by:

*p*_{1}=
(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)

*p*_{2}=
(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)

*s=*
(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)

*p*_{1} is a 5-fold rotation around the centre of
figure and
*p*_{2} 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 which evaluate the cycle index of a
permutation group given by a set of generators.
For instance the SYMMETRICA
routine `zykelind_arb(a,b)`

computes the cycle index `b`

of a
permutation group with generators given
in the `VECTOR`

-object `a`

.
In this way the cycle indices of *R*
and *S* acting on the set of
vertices are computed as

* Z(R,Vertices)=***(**1**)/(**60**)**(24 x_{5}^{12}
+20 x_{3}^{20}
+15 x_{2}^{30}
+x_{1}^{60})

* Z(S,Vertices)=***(**1**)/(**120**)**(24 x_{10}^{6}
+20 x_{6}^{10}
+24 x_{5}^{12}
+20 x_{3}^{20}
+16 x_{2}^{30}
+15 x_{1}^{4} x_{2}^{28}
+x_{1}^{60}).

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

| | | **The action on the set of vertices** |