3.2.3. Systematic numbering for (C₃₂-O_h)[4,6]fullerene

This fullerene features three symmetry-equivalent C₄ axes (Fig. 17a), four symmetry-equivalent C₃ axes (not shown: they pass through opposite atoms at the intersection of three six-membered rings), and six symmetry-equivalent C₂ axes (Fig. 17b). When one of the C₄ axes is used as the reference axis (Fig. 17a) there is only one possible pathway, and the numbering becomes discontiguous at position 28. Using one of the C₃ axes, the only possible pathway leads to a numbering discontiguous at position 25. Using one of the C₂ axes as the reference axis, there are three possible pathways that lead to numberings discontiguous at positions 25, 26, and 28, respectively. The latter is shown in Fig. 17b. Thus, two discontiguous numberings with a first segment of equal length are found. However, according to Fu-3.2.1b, the numbering reported in Fig. 17a is preferred, because its pathway uses the highest order axis as the reference axis. From position 28, the numbering resumes as shown in Fig. 17a (Fu-3.2.4a) to reach position 32 (Fu-3.2.4b) without further discontiguity.

Fig. 17. Systematic numbering of (C₃₂-O_h)[4,6]fullerene