ORIGIN CHOICE 1, Origin at -4, at -1/4, -1/4, -1/4 from -1
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
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| | |
| | (1) x, y, z | (2) -x, -y, z | (3) -y + 1/2, x + 1/2, z + 1/2 | (4) y + 1/2, -x + 1/2, z + 1/2 | | (5) -x + 1/2, -y + 1/2, -z + 1/2 | (6) x + 1/2, y + 1/2, -z + 1/2 | (7) y, -x, -z | (8) -y, x, -z |
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I Maximal translationengleiche subgroups
| [2] P-4 (81) | 1; 2; 7; 8 |
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| [2] P42 (77) | 1; 2; 3; 4 |
| 1/2, 0, 0
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| [2] P2/n (13, P112/a) | 1; 2; 5; 6 | -a - b, a, c
| 1/4, 1/4, 1/4
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II Maximal klassengleiche subgroups
[2] a' = 2a, b' = 2b, c' = 2c
| F41/d (88, I41/a) | <2; 3; 5> | a - b, a + b, 2c | |
| F41/d (88, I41/a) | <2; 3; 5 + (0, 0, 1)> | a - b, a + b, 2c | 0, 0, 1/2 |
| F41/d (88, I41/a) | <2; 5; 3 + (0, 0, 1)> | a - b, a + b, 2c | 1/2, 1/2, 0 |
| F41/d (88, I41/a) | <2; (3; 5) + (0, 0, 1)> | a - b, a + b, 2c | 1/2, 1/2, 1/2 |
[3] c' = 3c
 | P42/n (86) | <2; (3; 5) + (0, 0, 1)> | a, b, 3c | | | P42/n (86) | <2; 3 + (0, 0, 1); 5 + (0, 0, 3)> | a, b, 3c | 0, 0, 1 | | P42/n (86) | <2; 3 + (0, 0, 1); 5 + (0, 0, 5)> | a, b, 3c | 0, 0, 2 |
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- Series of maximal isomorphic subgroups
[p] c' = pc
| P42/n (86) | <2; 3 + (0, 0, p/2 - 1/2); 5 + (0, 0, p/2 - 1/2 + 2u)> | a, b, pc | 0, 0, u | | | p > 2; 0 ≤ u < p
p conjugate subgroups for the prime p |
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[p2] a' = pa, b' = pb
| P42/n (86) | <2 + (2u, 2v, 0); 3 + (p/2 - 1/2 + u + v, p/2 - 1/2 - u + v, 0); 5 + (p/2 - 1/2 + 2u, p/2 - 1/2 + 2v, 0)> | pa, pb, c | u, v, 0 | | | p > 2; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 3 (mod 4) |
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[p = q2 + r2] a' = qa - rb, b' = ra + qb
| P42/n (86) | <2 + (2u, 0, 0); 3 + (q/2 + r/2 - 1/2 + u, q/2 - r/2 - 1/2 - u, 0); 5 + (q/2 + r/2 - 1/2 + 2u, q/2 - r/2 - 1/2, 0)> | qa - rb, ra + qb, c | u, 0, 0 | | | q > 0; r > 0; p > 4; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 1 (mod 4) |
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I Minimal translationengleiche supergroups
| [2] P42/nbc (133); [2] P42/nnm (134); [2] P42/nmc (137); [2] P42/ncm (138) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [2] C42/m (84, P42/m); [2] I4/m (87) |
ORIGIN CHOICE 2, Origin at -1 on n, at 1/4, 1/4, 1/4 from -4
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
|
| | |
| | (1) x, y, z | (2) -x + 1/2, -y + 1/2, z | (3) -y, x + 1/2, z + 1/2 | (4) y + 1/2, -x, z + 1/2 | | (5) -x, -y, -z | (6) x + 1/2, y + 1/2, -z | (7) y, -x + 1/2, -z + 1/2 | (8) -y + 1/2, x, -z + 1/2 |
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I Maximal translationengleiche subgroups
| [2] P-4 (81) | 1; 2; 7; 8 |
| 1/4, 1/4, 1/4
|
| [2] P42 (77) | 1; 2; 3; 4 |
| 3/4, 1/4, 0
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| [2] P2/n (13, P112/a) | 1; 2; 5; 6 | -a - b, a, c
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II Maximal klassengleiche subgroups
[2] a' = 2a, b' = 2b, c' = 2c
| F41/d (88, I41/a) | <2; 3; 5> | a - b, a + b, 2c | |
| F41/d (88, I41/a) | <2; 3; 5 + (0, 0, 1)> | a - b, a + b, 2c | 0, 0, 1/2 |
| F41/d (88, I41/a) | <2; 5; 3 + (0, 0, 1)> | a - b, a + b, 2c | 1/2, 1/2, 0 |
| F41/d (88, I41/a) | <2; (3; 5) + (0, 0, 1)> | a - b, a + b, 2c | 1/2, 1/2, 1/2 |
[3] c' = 3c
| P42/n (86) | <2; 5; 3 + (0, 0, 1)> | a, b, 3c | | | P42/n (86) | <2; 3 + (0, 0, 1); 5 + (0, 0, 2)> | a, b, 3c | 0, 0, 1 | | P42/n (86) | <2; 3 + (0, 0, 1); 5 + (0, 0, 4)> | a, b, 3c | 0, 0, 2 |
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- Series of maximal isomorphic subgroups
[p] c' = pc
| P42/n (86) | <2; 3 + (0, 0, p/2 - 1/2); 5 + (0, 0, 2u)> | a, b, pc | 0, 0, u | | | p > 2; 0 ≤ u < p
p conjugate subgroups for the prime p |
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[p2] a' = pa, b' = pb
| P42/n (86) | <2 + (p/2 - 1/2 + 2u, p/2 - 1/2 + 2v, 0); 3 + (u + v, p/2 - 1/2 - u + v, 0); 5 + (2u, 2v, 0)> | pa, pb, c | u, v, 0 | | | p > 2; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 3 (mod 4) |
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[p = q2 + r2] a' = qa - rb, b' = ra + qb
| P42/n (86) | <2 + (q/2 + r/2 - 1/2 + 2u, q/2 - r/2 - 1/2, 0); 3 + (r/2 + u, q/2 - 1/2 - u, 0); 5 + (2u, 0, 0)> | qa - rb, ra + qb, c | u, 0, 0 | | | q > 0; q odd; r > 1; r even; p > 4; 0 ≤ u < p
p conjugate subgroups for each pair of q and r |
| P42/n (86) | <2 + (q/2 + r/2 + 1/2 + 2u, q/2 - r/2 - 1/2, 0); 3 + (r/2 + 1/2 + u, q/2 - 1 - u, 0); 5 + (1 + 2u, 0, 0)> | qa - rb, ra + qb, c | 1/2 + u, 0, 0 | | | q > 1; q even; r > 0; r odd; p > 4; 0 ≤ u < p
p conjugate subgroups for each pair of q and r |
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I Minimal translationengleiche supergroups
| [2] P42/nbc (133); [2] P42/nnm (134); [2] P42/nmc (137); [2] P42/ncm (138) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [2] C42/m (84, P42/m); [2] I4/m (87) |
