| R-3c | No. 167 | R-32/c | D3d6 |
HEXAGONAL AXES
| Axes | Coordinates | Wyckoff positions | ||||||
| 6a | 6b | 12c | 18d | 18e | 36f | |||
| I Maximal translationengleiche subgroups | ||||||||
| [2] R3c(161) | 6a | 6a | 2 × 6a | 18b | 18b | 2 × 18b | ||
| [2] R32 (155) | x, y, z + (1/4) | 3a; 3b | 6c | 2 × 6c | 18f | 9d; 9e | 2 × 18f | |
| [2] R-3 (148) | 6c | 3a; 3b | 2 × 6c | 9d; 9e | 18f | 2 × 18f | ||
| [3] C12/c1 | (1/3)(2a + b - 2c), | (2/3)x, -(1/2)x + y, x + z | 4e | 4a | 8f | 4b; 4c; 4d | 4e; 8f | 3 × 18f |
| (15) | b, c | |||||||
| conjugate: | (1/3)(-a + b - 2c), | (3/2)(-x + y), (1/2)(-x - y), | ||||||
| -a - b, c | -x + y + z | |||||||
| conjugate: | (1/3)(-a - 2b - 2c), | -(3/2) y, x - (1/2)y, -y + z | ||||||
| a, c | ||||||||
| alternative: | 2a + b, b, | (1/2)x + 2z, -(1/2)x + y, 3z | ||||||
| -(2/3)(2a + b) + (1/3)c | ||||||||
| or | -a + b, -a - b, | (1/2)(-x + y) + 2z, | ||||||
| (2/3)(a - b) + (1/3)c | -(1/2)(x + y), 3z | |||||||
| or | -a - 2b, a, | -(1/2)y + 2z, x - (1/2)y, 3z | ||||||
| (2/3)(a + 2b) + (1/3)c | ||||||||
| Loss of centring translations | ||||||||
| [3] P-3c1 (165) | 3 conjugate subgroups | 2a; 4d | 2b; 4d | 4c; 2 × 4d | 6e; 12g | 6f; 12g | 3 × 12g | |
| Enlarged unit cell, isomorphic | ||||||||
| [5] R-3c | -a, -b, 5c | -x, -y, (1/5)z; | 6a; 2 × 12c | 6b; 2 × 12c | 5 × 12c | 18d; 2 × 36f | 18e; 2 × 36f | 5 × 36 f |
| ±(0, 0, (1/5)); ±(0, 0, (2/5)) | ||||||||
| [p] R-3c | -a, -b, pc | -x, -y, (1/p)z; +(0, 0, (u/p)) | 6a; ((p - 1)/2) × 12c | 6b; ((p - 1)/2) × 12c | p × 12c | 18d; ((p - 1)/2) × 36f | 18e; ((p - 1)/2) × 36f | p × 36f |
| p = prime = 6n - 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] R-3c | -2a, -2b, c | -(1/2)x, -(1/2)y, z; +((1/2), 0, 0); | 6a; 18e | 6b; 18d | 12c; 36f | 2 × 36f | 2 × 18e; 36f | 4 × 36f |
| +(0, (1/2), 0); +((1/2), (1/2), 0) | ||||||||
| [p2] R-3c | -pa, -pb, c | -(1/p)x, -(1/p)y, z; +((u/p), (v/p), 0) | 6a; | 6b; ((p2 - 1)/6) × 36f | 12c; | 18d; ((p2 - 1)/2) × 36f | p × 18e; | p2 × 36f |
| p = prime = 6n - 1; u, v = 1, . . ., p - 1 | (p - 1) × 18e; | ((p2 - 1)/3) × 36f | ((p(p - 1))/2) × 36f | |||||
| pa, pb, c | (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) | (((p - 1)(p - 2))/6) × | ||||||
| 36f | ||||||||
| p = prime = 6n + 1; u, v = 1, . . ., p - 1 | ||||||||
