| Pc | No. 7 | Cs2 |
| P1c1 | P1n1 | P1a1 | |||||||
|
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 | |||||||||
| 2a | |||||||||
| I Maximal translationengleiche subgroups | |||||||||
| [2] P1 (1) | 2 × 1a | P1 | P1 | ||||||
| II Maximal klassengleiche subgroups | |||||||||
| Enlarged unit cell, non-isomorphic | |||||||||
| [2] C1c1 | 2a, 2b, c | (1/2)x, (1/2)y, z; +(0, (1/2), 0) | 2 × 4a | A1n1 | a - c, | x, (1/2)y, (1/2)(x + z); +(0, (1/2), 0) | I1a1 | a, 2b, | x - (1/2)z, (1/2)y, (1/2)z; +(0, (1/2), 0) |
| (9) | 2b, 2c | a + 2c | |||||||
| [2] C1c1 | 2a, 2b, c | (1/2)x, (1/2)y + (1/4), z; | 2 × 4a | A1n1 | a - c, | x, (1/2)y + (1/4), (1/2)(x + z); | I1a1 | a, 2b, | x - (1/2)z, (1/2)y + (1/4), (1/2)z; |
| (9) | +(0, (1/2), 0) | 2b, 2c | +(0, (1/2), 0) | a + 2c | +(0, (1/2), 0) | ||||
| Enlarged unit cell, isomorphic | |||||||||
| [2] P1c1 | a, 2b, c | x, (1/2)y, z; +(0, (1/2), 0) | 2 × 2a | P1n1 | a, 2b, c | x, (1/2)y, z; +(0, (1/2), 0) | P1a1 | a, 2b, c | x, (1/2)y, z; +(0, (1/2), 0) |
| [2] P1c1 | a, 2b, c | x, (1/2)y + (1/4), z; +(0, (1/2), 0) | 2 × 2a | P1n1 | a, 2b, c | x, (1/2)y + (1/4), z; +(0, (1/2), 0) | P1a1 | a, 2b, c | x, (1/2)y + (1/4), z; +(0, (1/2), 0) |
| [3] P1c1 | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | 3 × 2a | P1n1 | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | P1a1 | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) |
| [p] P1c1 | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) | p × 2a | P1n1 | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) | P1a1 | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) |
| p = prime; u = 1, . . ., p - 1 | p = prime; u = 1, . . ., p - 1 | p = prime; u = 1, . . ., p - 1 | |||||||
| [2] P1c1 | 2a, b, c | (1/2)x, y, z; +((1/2), 0, 0) | 2 × 2a | P1n1 | a - c, | x, y, (1/2)(x + z); +(0, 0, (1/2)) | P1a1 | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) |
| b, 2c | |||||||||
| [2] P1n1 | 2a, b, c | (1/2)x, y, z; +((1/2), 0, 0) | 2 × 2a | P1a1 | a - c, | x, y, (1/2)(x + z); +(0, 0, (1/2)) | P1n1 | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) |
| b, 2c | |||||||||
| [3] P1c1 | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | 3 × 2a | P1n1 | a - 2c, | x, y, (1/3)(2x + z); | P1a1 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) |
| b, 3c | ±(0, 0, (1/3)) | ||||||||
| [3] P1c1 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | 3 × 2a | P1n1 | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | P1a1 | 3a, b, a + c | (1/3)(x - z), y, z; ±((1/3), 0, 0) |
| [3] P1c1 | a + c, | x, y, (1/3)(-x + z); | 3 × 2a | P1n1 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | P1a1 | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) |
| b, 3c | ±(0, 0, (1/3)) | ||||||||
| [3] P1c1 | a - c, | x, y, (1/3)(x + z); | 3 × 2a | P1n1 | 2a + c, b, | (1/3)(2x - z), y, (1/3)(-x + 2z); | P1a1 | 3a, b, | (1/3)(x + z), y, z; |
| b, 3c | ±(0, 0, (1/3)) | a + 2c | ±((1/3), 0, (1/3)) | -a + c | ±((1/3), 0, 0) | ||||
| [p] P1c1 | pa, b, c | (1/p)x, y, z; +((u/p), 0, 0) | p × 2a | P1n1 | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | P1a1 | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) |
| p = prime; u = 1, . . ., p - 1; | p = prime; u = 1, . . ., p - 1; | p = prime; u = 1, . . ., p - 1; | |||||||
| [p] P1c1 | a + qc, | x, y, (1/p)(-qx + z); | p × 2a | P1n1 | pa, b, | (1/p)(x - 2qz), y, z; | P1a1 | pa, b, | (1/p)(x - qz), y, z; |
| b, pc | +(0, 0, (u/p)) | 2qa + c | +((u/p), 0, 0) | qa + c | +((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
| P11a | P11n | P11b | |||||||
|
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 (1) | 2 × 1a | P1 | P1 | ||||||
| II Maximal klassengleiche subgroups | |||||||||
| Enlarged unit cell, non-isomorphic | |||||||||
| [2] A11a | a, 2b, 2c | x, (1/2)y, (1/2)z; +(0, 0, (1/2)) | 2 × 4a | B11n | 2a, | (1/2)(x + y), y, (1/2)z; +(0, 0, (1/2)) | I11b | 2a + b, | (1/2)x, -(1/2)x + y, (1/2)z; +(0, 0, (1/2)) |
| (9) | -a + b, 2c | b, 2c | |||||||
| [2] A11a | a, 2b, 2c | x, (1/2)y, (1/2)z + (1/4); | 2 × 4a | B11n | 2a, | (1/2)(x + y), y, (1/2)z + (1/4); | I11b | 2a + b, | (1/2)x, -(1/2)x + y, (1/2)z + (1/4); |
| (9) | +(0, 0, (1/2)) | -a + b, 2c | +(0, 0, (1/2)) | b, 2c | +(0, 0, (1/2)) | ||||
| Enlarged unit cell, isomorphic | |||||||||
| [2] P11a | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) | 2 × 2a | P11n | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) | P11b | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) |
| [2] P11a | a, b, 2c | x, y, (1/2)z + (1/4); +(0, 0, (1/2)) | 2 × 2a | P11n | a, b, 2c | x, y, (1/2)z + (1/4); +(0, 0, (1/2)) | P11b | a, b, 2c | x, y, (1/2)z + (1/4); +(0, 0, (1/2)) |
| [3] P11a | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | 3 × 2a | P11n | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | P11b | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) |
| [p] P11a | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | p × 2a | P11n | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) | P11b | a, b, pc | x, y, (1/p)z; +(0, 0, (u/p)) |
| p = prime; u = 1, . . ., p - 1 | p = prime; u = 1, . . ., p - 1 | p = prime; u = 1, . . ., p - 1 | |||||||
| [2] P11a | a, 2b, c | x, (1/2)y, z; +(0, (1/2), 0) | 2 × 2a | P11n | 2a, | (1/2)(x + y), y, z; +((1/2), 0, 0) | P11b | 2a, b, c | (1/2)x, y, z; +((1/2), 0, 0) |
| -a+b, c | |||||||||
| [2] P11n | a, 2b, c | x, (1/2)y, z; +(0, (1/2), 0) | 2 × 2a | P11b | 2a, | (1/2)(x + y), y, z; +((1/2), 0, 0) | P11n | 2a, b, c | (1/2)x, y, z; +((1/2), 0, 0) |
| -a + b, c | |||||||||
| [3] P11a | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | 3 × 2a | P11n | 3a, | (1/3)(x + 2y), y, z; | P11b | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) |
| -2a + b, c | ±((1/3), 0, 0) | ||||||||
| [3] P11a | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | 3 × 2a | P11n | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) | P11b | a + b, 3b, c | x, (1/3)(-x + y), z; ±(0, (1/3), 0) |
| [3] P11a | 3a, | (1/3)(x - y), y, z; | 3 × 2a | P11n | 3a, b, c | (1/3)x, y, z; ±((1/3), 0, 0) | P11b | a, 3b, c | x, (1/3)y, z; ±(0, (1/3), 0) |
| a + b, c | ±((1/3), 0, 0) | ||||||||
| [3] P11a | 3a, | (1/3)(x + y), y, z; | 3 × 2a | P11n | 2a + b, | (1/3)(2x - y), (1/3)(-x + 2y), z; | P11b | a - b | x, (1/3)(x + y), z; |
| -a + b, c | ±((1/3), 0, 0) | a + 2b, c | ±((1/3), (1/3), 0) | 3b, c | ±(0, (1/3), 0) | ||||
| [p] P11a | a, pb, c | x, (1/p)y, z; +(0, (u/p), 0) | p × 2a | P11n | pa, b, c | (1/p)x, y, z; +((u/p), 0, 0) | P11b | pa, b, c | (1/p)x, y, z; +((u/p), 0, 0) |
| p = prime; u = 1, . . ., p - 1; | p = prime; u = 1, . . ., p - 1; | p = prime; u = 1, . . ., p - 1; | |||||||
| [p] P11a | pa, | (1/p)(x - qy), y, z; | p × 2a | P11n | a + 2qb, | x, (1/p)(-2qx + y), z; | P11b | a + qb, | x, (1/p)(-qx + y), z; |
| qa + b, c | +((u/p), 0, 0) | pb, c | +(0, (u/p), 0) | pb, c | +(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; | |||||||
| -(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
