Figure 1.

Subgroup relationships with indices 2 (solid lines) and 3 (dashed line) are shown in (a). All other relationships can be inferred by identifying the shortest path through the hierarchy, and multiplying the subgroup indices. For example, P1 is related to P6 through P6 → P3 (index 2) and P3 → P1 (index 3) so P1 is a subgroup of P6 with index 3 × 2 = 6. We also show enlarged versions of some of the subgroup relationships involving (b) P6 and (c) PMM, and highlight the symmetries within the subgroups to emphasize how the supergroup can be generated by adding additional transformations to the subgroup. Illustration adapted from Wade [30].