| R32 | No. 155 | D37 |
HEXAGONAL AXES
| Axes | Coordinates | Wyckoff positions | ||||||
| 3a | 3b | 6c | 9d | 9e | 18f | |||
| I Maximal translationengleiche subgroups | ||||||||
| [2] R3 (146) | 3a | 3a | 2 × 3a | 9b | 9b | 2 × 9b | ||
| [3] C121 | (1/3)(2a + b - 2c), b, | (1/2)x - z, -(1/2)x + y, x + z | 2a | 2b | 4c | 2a; 4c | 2b; 4c | 3 × 4c |
| (5) | (1/3)(2a + b + c) | |||||||
| conjugate: | (1/3)(-a + b - 2c), | -(1/2)x + (1/2)y - z, -(1/2)(x + 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 | ||||||
| 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 | ||||||
| -(1/3)(2a + b + c) | ||||||||
| or | -a + b, a + b, | (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] P3221 | x + (1/3), y + (1/3), z | 3a | 3b | 6c | 3a; 6c | 3b; 6c | 3 × 6c | |
| (154) | 3 conjugate subgroups | |||||||
| [3] P3121 | x - (1/3), y - (1/3), z | 3a | 3b | 6c | 3a; 6c | 3b; 6c | 3 × 6c | |
| (152) | 3 conjugate subgroups | |||||||
| [3] P321 | 3 conjugate subgroups | 1a; 2d | 1b; 2d | 2c; 2 × 2d | 3e; 6g | 3f; 6g | 3 × 6g | |
| (150) | ||||||||
| Enlarged unit cell, isomorphic | ||||||||
| [2] R32 | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 3a; 3b | 6c | 2 × 6c | 9d; 9e | 18f | 2 × 18f |
| [2] R32 | -a, -b, 2c | -x, -y, (1/2)z + (1/4); +(0, 0, (1/2)) | 6c | 3a; 3b | 2 × 6c | 18f | 9d; 9e | 2 × 18f |
| [5] R32 | -a, -b, 5c | -x, -y, (1/5)z; ±(0, 0, (1/5)) | 3a; 2 × 6c | 3b; 2 × 6c | 5 × 6c | 9d; 2 × 18f | 9e; 2 × 18f | 5 × 18f |
| ±(0, 0, (2/5)) | ||||||||
| [p] R32 | -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) × | 9e; ((p - 1)/2) × | p × 18f |
| p = prime = 6n - 1; u = 1, . . ., p - 1 | 18f | 18f | ||||||
| a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | |||||||
| p = prime = 6n + 1; u = 1, . . ., p - 1 | ||||||||
| [4] R32 | -2a, -2b, c | -(1/2)x, -(1/2)y, z; +((1/2), 0, 0); | 3a; 9d | 3b; 9e | 6c; 18f | 2 × 9d; 18f | 2 × 9e; 18f | 4 × 18f |
| +(0, (1/2), 0); +((1/2), (1/2), 0) | ||||||||
| [p2] R32 | -pa, -pb, c | - (1/p)x, -(1/p)y, z; +((u/p), (v/p), 0) | 3a; (p - 1) × 9d; | 3b; (p - 1) × 9e; | 6c; | p × 9d; | p × 9e; | p2 × 18f |
| p = prime = 6n - 1; u, v = 1, . . ., p - 1 | (((p - 1)(p - 2))/6) × | (((p - 1)(p - 2))/6) × | ((p2 - 1)/3) × 18f | ((p(p - 1))/2) × | ((p(p - 1))/2) × | |||
| 18f | 18f | 18f | 18f | |||||
| pa, pb, c | (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) | |||||||
| p = prime = 6n + 1; u, v = 1, . . ., p - 1 | ||||||||
