| R3 | No. 146 | C34 |
HEXAGONAL AXES
| Axes | Coordinates | Wyckoff positions | |||
| 3a | 9b | ||||
| I Maximal translationengleiche subgroups | |||||
| [3] P1 (1) | (1/3)(2a + b + c), (1/3)(-a + b + c), | x + z, -x + y + z, -y + z | 1a | 3 × 1a | |
| (1/3)(-a - 2b + c) | |||||
| rhombohedral axes: a, b, c | x, y, z | ||||
| II Maximal klassengleiche subgroups | |||||
| Loss of centring translations | |||||
| [3] P32 (145) | x - (1/3), y, z | 3a | 3 × 3a | ||
| [3] P31 (144) | x + (1/3), y, z | 3a | 3 × 3a | ||
| [3] P3 (143) | 1a; 1b; 1c | 3 × 3d | |||
| Enlarged unit cell, isomorphic | |||||
| [2] R3 | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 2 × 3a | 2 × 9b | |
| [p] R3 | -a, -b, pc | -x, -y, (1/p)z; +(0, 0, (u/p)) | p × 3a | p × 9b | |
| p = prime = 3n - 1; u = 1, . . ., p - 1 | |||||
| [p] R3 | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | p × 3a | p × 9b | |
| p = prime = 6n + 1; u = 1, . . ., p - 1 | |||||
| [7] R3 | 3a + b, -a + 2b, c | (1/7)(2x + y), (1/7)(-x + 3y), z; | 3a; 2 × 9b | 7 × 9b | |
| ±((1/7), (3/7), 0); ±((5/7), (1/7), 0); ±((3/7), (2/7), 0) | |||||
| [7] R3 | a + 3b, -3a - 2b, c | (1/7)(-2x + 3y), (1/7)(-3x + y), z; | 3a; 2 × 9b | 7 × 9b | |
| ±((3/7), (1/7), 0); ±((1/7), (5/7), 0); ±((2/7), (3/7), 0) | |||||
| [p] R3 | qa + rb, -ra +(q - r)b, c | (1/p)((q - r)x + ry), (1/p)(-rx + qy), z; | 3a; (1/3)(p - 1) × 9b | p × 9b | |
| +((ur/p), (uq/p), 0) | |||||
| p = q2 - qr + r2 = prime = 6n + 1; | |||||
| q, r = 1, 2, . . .; q≠ r; q + r = 3n + 1; u = 1, . . ., p - 1 | |||||
| [4] R3 | -2a, -2b, c | -(1/2)x, -(1/2)y, z; +((1/2), 0, 0); | 3a; 9b | 4 × 9b | |
| +(0, (1/2), 0); +((1/2), (1/2), 0) | |||||
| [p2] R3 | -pa, -pb, c | -(1/p)x, -(1/p)y, z; +((u/p), (v/p), 0) | 3a; (1/3)(p2 - 1) × 9b | p2 × 9b | |
| p = prime = 3n - 1; u, v = 1, . . ., p - 1 | |||||
