| R-3m | No. 166 | R-32/m | D3d5 |
HEXAGONAL AXES
| Axes | Coordinates | Wyckoff positions | |||||
| 3a | 3b | 6c | 9d | 9e | |||
| 18f | 18g | 18h | 36i | ||||
| I Maximal translationengleiche subgroups | |||||||
| [2] R3m (160) | 3a | 3a | 2 × 3a | 9b | 9b | ||
| 18c | 18c | 2 × 9b | 2 × 18c | ||||
| [2] R32 (155) | 3a | 3b | 6c | 9e | 9d | ||
| 2 × 9d | 2 × 9e | 18f | 2 × 18f | ||||
| [2] R-3 (148) | 3a | 3b | 6c | 9d | 9e | ||
| 18f | 18f | 18f | 2 × 18f | ||||
| [3] C12/m1 | (1/3)(2a + b - 2c), b, | (1/2)x - z, -(1/2)x + y, x + z | 2a | 2d | 4i | 2c; 4e | 2b; 4f |
| (12) | (1/3)(2a + b + c) | 4g; 8j | 4h; 8j | 4i; 8j | 3 × 8j | ||
| conjugate: | (1/3)(-a + b - 2c), | -(1/2)x + (1/2)y - z, -(1/2)x - (1/2)y, | |||||
| -a - b, (1/3)(-a + b + c) | -x + y + z | ||||||
| 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 | 2a | 2c | 4i | 2d; 4e | 2b; 4f |
| 4g; 8j | 4h; 8j | 4i; 8j | 3 × 8j | ||||
| or | (1/3)(-a + b - 2c), | (3/2)(-x + y), -(1/2)(x + y), | |||||
| -a - b, c | -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 | 2a | 2d | 4i | 2c; 4f | 2b; 4e |
| -(1/3)(2a + b + c) | 4g; 8j | 4h; 8j | 4i; 8j | 3 × 8j | |||
| or | -a + b, a + b, (1/3)(a - b - c) | (1/2)(-x + y) - z, (1/2)(x + y), -3z | |||||
| or | -a - 2b, -a, (1/3)(a + 2b - c) | -(1/2)y - z, -x + (1/2)y, -3z | |||||
| II Maximal klassengleiche subgroups | |||||||
| Loss of centring translations | |||||||
| [3] P-3m1 | 1a; 2d | 1b; 2d | 2c; 2 × 2d | 3f; 6i | 3e; 6i | ||
| (164) | 3 conjugate subgroups | 6g; 12j | 6h; 12j | 3 × 6i | 3 × 12j | ||
| Enlarged unit cell, non-isomorphic | |||||||
| [2] R-3c | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 6b | 6a | 12c | 18e | 18d |
| (167) | 36f | 2 × 18e | 36f | 2 × 36f | |||
| [2] R-3c | -a, -b, 2c | -x, -y, (1/2)z + (1/4); +(0, 0, (1/2)) | 6a | 6b | 12c | 18d | 18e |
| (167) | 2 × 18e | 36f | 36f | 2 × 36f | |||
| Enlarged unit cell, isomorphic | |||||||
| [2] R-3m | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 3a; 3b | 6c | 2 × 6c | 18h | 9d; 9e |
| 18 f; 18g | 36i | 2 × 18h | 2 × 36i | ||||
| [2] R-3m | -a, -b, 2c | -x, -y, (1/2)z + (1/4); +(0, 0, (1/2)) | 6c | 3a; 3b | 2 × 6c | 9d; 9e | 18h |
| 36i | 18f; 18g | 2 × 18h | 2 × 36i | ||||
| [p] R-3m | -a, -b, pc | -x, -y, (1/p)z; +(0, 0, (u/p)) | 3a; ((p - 1)/2) × 6c | 3b; ((p - 1)/2) × 6c | p × 6c | 9d; ((p - 1)/2) × 18h | 9e; ((p - 1)/2) × 18h |
| p = prime = 6n - 1; u = 1, . . ., p - 1 | 18f; ((p - 1)/2) × 36i | 18g; ((p - 1)/2) × 36i | p × 18h | p × 36i | |||
| a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | ||||||
| p = prime = 6n + 1; u = 1, . . ., p - 1 | |||||||
| [4] R-3m | -2a, -2b, c | -(1/2)x, -(1/2)y, z; +((1/2), 0, 0); | 3a; 9e | 3b; 9d | 6c; 18h | 18g; 18h | 18f; 18h |
| +(0, (1/2), 0); +((1/2), (1/2), 0) | 2 × 18f; 36i | 2 × 18g; 36i | 2 × 18h; 36i | 4 × 36i | |||
| [p2] R-3m | -pa, -pb, c | -(1/p)x, -(1/p)y, z; +((u/p), (v/p), 0) | 3a; ((p - 1)/2) × 18f; | 3b; ((p - 1)/2) × 18g; | 6c; | 9d; ((p - 1)/2) × 18g; | 9e; ((p - 1)/2) × 18f; |
| p = prime = 6n - 1; u, v = 1, . . ., p - 1 | ((p - 1)/2) × 18h; | ((p - 1)/2) × 18h; | (p - 1) × 18h; | ((p - 1)/2) × 18h; | ((p - 1)/2) × 18h; | ||
| pa, pb, c | (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) | (((p - 1)(p - 5))/12) × 36i | (((p - 1)(p - 5))/12) × 36i | (((p - 1)(p - 2))/6) × 36i | (((p - 1)2)/4) × 36i | (((p - 1)2)/4) × 36i | |
| p = prime = 6n + 1; u, v = 1, . . ., p - 1 | p × 18f; | p × 18g; | p × 18h; | p2 × 36i | |||
| ((p(p - 1))/2) × 36i | ((p(p - 1))/2) × 36i | ((p(p - 1))/2) × 36i | |||||
