This fullerene features six symmetry-equivalent C5 axes (connecting midpoints of opposite five-membered rings) ten symmetry-equivalent C3 axes (connecting atoms at the intersection of three six-membered rings), and fifteen symmetry-equivalent C2 axes (passing through the midpoints of opposite hexagons). These axes are shown in Fig. 21a. When one of the C5 axes is used as the reference axis, the pathway can begin at any one of the five symmetry-equivalent atoms of the pentagon (indicated as a in Fig. 21a); the numbering (shown in Fig. 21b) becomes discontiguous at position 75. When one of the C3 axes is used as the reference axis only one pathway from atom b must be considered; this pathway becomes discontiguous at position 66, and is therefore discarded (Fu-3.2.3). When one of the C2 axes is used as the reference axis there are three pathways to be considered: c to d to e; d to e to c'; e to c' to d'. One of these leads to the numbering becoming discontiguous at position 75 (not shown); the others lead to numberings in which the discontiguities occur at positions 72 and 66. According to Fu-3.2.1 the three numberings built around the C2 axis are discarded. The correct numbering is therefore that shown in Fig. 21b. The completion of the numbering by application of Fu-3.2.4 is straightforward.
Fig. 21. Systematic numbering of (C80-Ih)[5,6]fullerene
5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995