| Cc | No. 9 | Cs4 |
| C1c1 | A1n1 | I1a1 | |||||||
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UNIQUE AXIS b |
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CELL CHOICE 1 |
CELL CHOICE 2 |
CELL CHOICE 3 |
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| Axes | Coordinates | Wyckoff | Axes | Coordinates | Axes | Coordinates | |||
| positions | |||||||||
| 4a | |||||||||
| I Maximal translationengleiche subgroups | |||||||||
| [2] P1 (1) | (1/2)(a - b), | 2x, x + y, z | 2 × 2a | P1 | a, b, | x, x + y, 2z | P1 | a, | x + y, 2y, y + z |
| b, c | (1/2)(-b + c) | (1/2)(-a + b - c), c | |||||||
| II Maximal klassengleiche subgroups | |||||||||
| Loss of centring translations | |||||||||
| [2] P1c1 (7) | 2 × 2a | P1n1 | P1a1 | ||||||
| [2] P1n1 (7) | x, y + (1/4), z | 2 × 2a | P1a1 | x, y + (1/4), z | P1c1 | x, y + (1/4), z | |||
| Enlarged unit cell, isomorphic | |||||||||
| [3] C1c1 | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | 3 × 4a | A1n1 | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | I1a1 | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) |
| [p] C1c1 | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) | p × 4a | A1n1 | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) | I1a1 | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) |
| p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | |||||||
| [3] C1c1 | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | 3 × 4a | A1n1 | a - 2c, b, 3c | x, y, (1/3)(2x + z); | I1a1 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) |
| ±(0, 0, (1/3)) | |||||||||
| [3] C1c1 | 3a, b, 2a + c | (1/3)(x - 2z), y, z; | 3 × 4a | A1n1 | a + 2c, b, 3c | x, y, (1/3)(-2x + z); | I1a1 | 3a, b, 2a + c | (1/3)(x - 2z), y, z; |
| ±((1/3), 0, 0) | ±(0, 0, (1/3)) | ±((1/3), 0, 0) | |||||||
| or: | 3a, b, -a + c | (1/3)(x + z), y + (1/4), z; | |||||||
| ±((1/3), 0, 0) | |||||||||
| [3] C1c1 | 3a, b, | (1/3)(x + 2z), y, z; | 3 × 4a | A1n1 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | I1a1 | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) |
| -2a + c | ±((1/3), 0, 0) | ||||||||
| or: | 3a, b, a + c | (1/3)(x - z), y + (1/4), z; | |||||||
| ±((1/3), 0, 0) | |||||||||
| [3] C1c1 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | 3 × 4a | A1n1 | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | I1a1 | 3a, b, -2a + c | (1/3)(x + 2z), y, z; |
| ±((1/3), 0, 0) | |||||||||
| [p] C1c1 | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | p × 4a | A1n1 | pa, b, c | (1/p)x, y, z; +((u/p), 0, 0) | I1a1 | pa, b, c | (1/p)x, y, z; +((u/p), 0, 0) |
| p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | |||||||
| [p] C1c1 | pa, b, | (1/p)(x - 2qz), y, z; | p × 4a | A1n1 | a + 2qc, b, pc | x, y, (1/p)(-2qx + z); | I1a1 | a + 2qc, b, pc | x, y, (1/p)(-2qx + z); |
| 2qa + c | +((u/p), 0, 0) | +(0, 0, (u/p)) | +(0, 0, (u/p)) | ||||||
| p = prime > 2; u = 1, . . ., p - 1; | p = prime > 2; u = 1, . . ., p - 1; | p = prime > 2; u = 1, . . ., p - 1; | |||||||
| -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1) | -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1) | -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1) | |||||||
Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3
| A11a | B11n | I11b | |||||||
|
UNIQUE AXIS c |
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|
CELL CHOICE 1 |
CELL CHOICE 2 |
CELL CHOICE 3 |
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| Axes | Coordinates | Wyckoff | Axes | Coordinates | Axes | Coordinates | |||
| positions | |||||||||
| 2a | |||||||||
| I Maximal translationengleiche subgroups | |||||||||
| [2] P1 | a, | x, 2y, y + z | 2 × 2a | P1 | (1/2)(a - c), | 2x, y, y + z | P1 | a, b, | x + y, y + z, 2z |
| (1/2)(b - c), c | b, c | (1/2)(-a - b + c) | |||||||
| II Maximal klassengleiche subgroups | |||||||||
| Loss of centring translations | |||||||||
| [2] P11a | 2 × 2a | P11n | P11b | ||||||
| [2] P11n | x, y, z + (1/4) | 2 × 2a | P11b | x, y, z + (1/4) | P11a | x, y, z + (1/4) | |||
| Enlarged unit cell, isomorphic | |||||||||
| [3] A11a | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | 3 × 4a | B11n | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | I11b | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) |
| [p] A11a | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | p × 4a | B11n | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | I11b | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) |
| p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | |||||||
| [3] A11a | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | 3 × 4a | B11n | 3a, -2a + b, c | (1/3)(x + 2y), y, z; | I11b | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) |
| ±((1/3), 0, 0) | |||||||||
| [3] A11a | a + 2b, 3b, c | x, (1/3)(-2x + y), z; | 3 × 4a | B11n | 3a, 2a + b, c | (1/3)(x - 2y), y, z; | I11b | a + 2b, 3b, c | x, (1/3)(-2x + y), z; |
| ±(0, (1/3), 0) | ±((1/3), 0, 0) | ±(0, (1/3), 0) | |||||||
| or: | a - b, 3b, c | x, (1/3)(x + y), z + (1/4); | |||||||
| ±(0, (1/3), 0) | |||||||||
| [3] A11a | a - 2b, | x, (1/3)(2x + y), z; | 3 × 4a | B11n | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | I11b | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) |
| 3b, c | ±(0, (1/3), 0) | ||||||||
| or: | a + b, 3b, c | x, (1/3)(-z + y), z + (1/4); | |||||||
| ±(0, (1/3), 0) | |||||||||
| [3] A11a | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | 3 × 4a | B11n | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | I11b | a - 2b, 3b, c | x, (1/3)(2x + y), z; |
| ±(0, (1/3), 0) | |||||||||
| [p] A11a | pa, b, c | (1/p)x, y, z; +((u/p), 0, 0) | p × 4a | B11n | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) | I11b | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) |
| p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | p = prime > 2; u = 1, . . ., p - 1 | |||||||
| [p] A11a | a + 2qb, | x, (1/p)(-2qx + y), z; | p × 4a | B11n | pa, 2qa + b, c | (1/p)(x - 2qy), y, z; | I11b | pa, 2qa + b, c | (1/p)(x - 2qy), y, z; |
| pb, c | +(0, (u/p), 0) | +((u/p), 0, 0) | +((u/p), 0, 0) | ||||||
| p = prime > 2; u = 1, . . ., p - 1; | p = prime > 2; u = 1, . . ., p - 1; | p = prime > 2; u = 1, . . ., p - 1; | |||||||
| -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1) | -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1) | -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1) | |||||||
Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3
