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All index 2 containments between groups with different parameters n and m = n2 are indicated by the lighter lines beside the boxes. Thus, ∗22n contains both ∗22m and 2∗m (each twice because there are two lines). 2. The groups classified by order. All the groups of a given order appear on one line, preceded by their number, with the number of isomorphism types as a subscript. The “rivers” separate groups of distinct structures (as indicated below the table). The lines for orders 8n, 8n + 1, . .

The finite groups of quaternions are 2I 2O 2T 2D2n 2Cn 1Cn (n odd). 6 Chiral and Achiral, Diploid and Haploid The word “chiral” (meaning “handed”) was introduced to science by Lord Kelvin before 1896 to denote objects that cannot be moved into coincidence with their mirror images. For the opposite concept, the word “achiral” (“unhanded”) or “amphichiral” (“either-handed”) has been used. We shall use these terms not only for such objects, but also for their symmetry groups. 1 For a finite physical object, all symmetries fix the center of gravity, and, when we take this to be the origin, are represented by orthogonal matrices whose determinants are necessarily ±1.

Appendix: Completeness of the Tables 53 while for ±[D × D(s) ] · 2 we have Variables s (mod f ) Conditions 2 s = fg − 1 Equalities s ≈ −s . 3. In the last eight lines, it is always permissible to replace D2 by C2 and D4 by D4 . 4. Different achiral groups.