| R3m | No. 160 | C3v5 |
HEXAGONAL AXES
| Axes | Coordinates | Wyckoff positions | ||||
| 3a | 9b | 18c | ||||
| I Maximal translationengleiche subgroups | ||||||
| [2] R3 (146) | 3a | 9b | 2 × 9b | |||
| [3] C1m1 (8) | (1/3)(2a + b - 2c), b, | (1/2)x - z, -(1/2)x + y, x + z | 2a | 2a; 4b | 3 × 4b | |
| (1/3)(2a + b + c) | ||||||
| conjugate: | (1/3)(-a + b - 2c), -a - b, | -(1/2)x + (1/2)y - z, -(1/2)x - (1/2)y, -x + y + z | ||||
| (1/3)(-a + b + c) | ||||||
| conjugate: | (1/3)(-a - 2b - 2c), a, | -(1/2)y - z, x - (1/2)y, -y + z | ||||
| (1/3)(-a - 2b + c) | ||||||
| alternative: | (1/3)(2a + b - 2c), b, c | (3/2)x, -(1/2)x + y, x + z | ||||
| or | (1/3)(-a + b - 2c), -a - b, c | (3/2)(-x + y), -(1/2)(x + y), -x + y + z | ||||
| or | (1/3)(-a - 2b - 2c), a, c | -(3/2)y, x - (1/2)y, -y + z | ||||
| alternative: | 2a + b, -b, | (1/2)x - z, (1/2)x - y, -3z | ||||
| -(1/3)(2a + b + c) | ||||||
| or | -a + b, a + b, (1/3)(a - b - c) | (1/2)(-x + y) - z, (1/2)(x + y), -3z | ||||
| (1/3)(a - b - c) | ||||||
| or | -a - 2b, -a, | -(1/2)y - z, -x + (1/2)y, -3z | ||||
| (1/3)(a + 2b - c) | ||||||
| II Maximal klassengleiche subgroups | ||||||
| Loss of centring translations | ||||||
| [3] P3m1 (156) | 1a; 1b; 1c | 3 × 3d | 3 × 6e | |||
| Enlarged unit cell, non-isomorphic | ||||||
| [2] R3c (161) | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 6a | 18b | 2 × 18b | |
| Enlarged unit cell, isomorphic | ||||||
| [2] R3m | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 2 × 3a | 2 × 9b | 2 × 18c | |
| [p] R3m | -a, -b, pc | -x, -y, (1/p)z; +(0, 0, (u/p)) | p × 3a | p × 9b | p × 18c | |
| p = prime = 3n - 1; u = 1, . . ., p - 1 | ||||||
| a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | |||||
| p = prime = 6n + 1; u = 1, . . ., p - 1 | ||||||
| [4] R3m | -2a, -2b, c | -(1/2)x, -(1/2)y, z; +((1/2), 0, 0); | 3a; 9b | 2 × 9b; 18c | 4 × 18c | |
| +(0, (1/2), 0); +((1/2), (1/2), 0) | ||||||
| [p2] R3m | -pa, -pb, c | -(1/p)x, -(1/p)y, z; +((u/p), (v/p), 0) | 3a; (p - 1) × 9b; | p × 9b; | p2 × 18c | |
| p = prime = 3n - 1; u, v = 1, . . ., p - 1 | (((p - 1)(p - 2))/6) × 18c | ((p(p - 1))/2) × 18c | ||||
| pa, pb, c | (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) | |||||
| p = prime = 6n + 1; u, v = 1, . . ., p - 1 | ||||||
