| R-3 | No. 148 | C3i2 |
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] P-1 | (1/3)(2a + b + c), | x + z, -x + y + z, -y + z | 1a | 1h | 2i | 1b; 1c; 1d | 1e; 1f; 1g | 3 × 2i |
| (2) | (1/3)(-a + b + c), | |||||||
| (1/3)(-a - 2b + c) | ||||||||
| rhombohedral axes: a, b, c | x, y, z | |||||||
| II Maximal klassengleiche subgroups | ||||||||
| Loss of centring translations | ||||||||
| [3] P-3(147) | 1a; 2d | 1b; 2d | 2c; 2 × 2d | 3f; 6g | 3e; 6g | 3 × 6g | ||
| Enlarged unit cell, isomorphic | ||||||||
| [2] R-3 | -a, -b, 2c | -x, -y, (1/2)z; +(0, 0, (1/2)) | 3a; 3b | 6c | 2 × 6c | 18f | 9d; 9e | 2 × 18f |
| [2] R-3 | -a, -b, 2c | -x, -y, (1/2)z + (1/4); +(0, 0, (1/2)) | 6c | 3a; 3b | 2 × 6c | 9d; 9e | 18f | 2 × 18f |
| [p] R-3 | -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) × 18f | 9e; ((p - 1)/2) × 18f | p × 18f |
| 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 | ||||||||
| [7] R-3 | 3a+b, | (1/7)(2x + y), (1/7)(-x + 3y), z; | 3a; 18f | 3b; 18f | 6c; 2 × 18f | 9d; 3 × 18f | 9e; 3 × 18f | 7 × 18f |
| -a + 2b, c | ±((1/7), (3/7), 0); ±((5/7), (1/7), 0); ±((3/7), (2/7), 0) | |||||||
| [7] R-3 | a+3b, | (1/7)(-2x + 3y), (1/7)(-3x + y), z; | 3a; 18f | 3b; 18f | 6c; 2 × 18f | 9d; 3 × 18f | 9e; 3 × 18f | 7 × 18f |
| -3a - 2b, c | ±((1/7), (5/7), 0); ±((2/7), (3/7), 0); ±((3/7), (1/7), 0) | |||||||
| [p] R-3 | qa + rb, | (1/p)((q - r)x + ry), | 3a; ((p - 1)/6) × 18f | 3b; ((p - 1)/6) × 18f | 6c; ((p - 1)/3) × 18f | 9d; ((p - 1)/2) × 18f | 9e; ((p - 1)/2) × 18f | p × 18f |
| -ra +(q - r)b, c | (1/p)(-rx + qy), z; +((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] R-3 | -2a, -2b, c | -(1/2)x, -(1/2)y, z; +((1/2), 0, 0); | 3a; 9e | 3b; 9d | 6c; 18f | 2 × 18f | 2 × 18f | 4 × 18f |
| +(0, (1/2), 0); +((1/2), (1/2), 0) | ||||||||
| [p2] R-3 | -pa, -pb, c | -(1/p)x, -(1/p)y, z; +((u/p), (v/p), 0) | 3a; ((p2 - 1)/6) × 18f | 3b; ((p2 - 1)/6) × 18f | 6c; ((p2 - 1)/3) × 18f | 9d; ((p2 - 1)/2) × 18f | 9e; ((p2 - 1)/2) × 18f | p2 × 18f |
| p = prime = 6n - 1; u, v = 1, . . ., p - 1 | ||||||||
