HEXAGONAL AXES
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(2/3, 1/3, 1/3); (2); (4)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
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| | (0, 0, 0)+ (2/3, 1/3, 1/3)+ (1/3, 2/3, 2/3)+ |
| | (1) x, y, z | (2) -y, x - y, z | (3) -x + y, -x, z | | (4) -y, -x, z | (5) -x + y, y, z | (6) x, x - y, z |
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I Maximal translationengleiche subgroups
| [2] R31 (146, R3) | (1; 2; 3)+ |
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 | [3] R1m (8, C1m1) | (1; 4)+ | 1/3(-a + b - 2c), -a - b, c
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| | [3] R1m (8, C1m1) | (1; 5)+ | 1/3(-a - 2b - 2c), a, c
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| | [3] R1m (8, C1m1) | (1; 6)+ | 1/3(2a + b - 2c), b, c
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II Maximal klassengleiche subgroups
- Loss of centring translations
| [3] P3m1 (156) | 1; 2; 3; 4; 5; 6 | | |
[2] a' = -b, b' = a + b, c' = 2c
| R3m (160) | <2; 4> | -b, a + b, 2c | |
[2] a' = a + b, b' = -a, c' = 2c
| R3c (161) | <2; 4 + (0, 0, 1)> | a + b, -a, 2c | |
[4] a' = -2b, b' = 2a + 2b
 | R3m (160) | <2; 4> | -2b, 2a + 2b, c | | | R3m (160) | <2 + (1, -1, 0); 4 + (1, 1, 0)> | -2b, 2a + 2b, c | 1, 0, 0 | | R3m (160) | <2 + (1, 2, 0); 4 + (1, 1, 0)> | -2b, 2a + 2b, c | 0, 1, 0 | | R3m (160) | <2 + (2, 1, 0); 4 + (2, 2, 0)> | -2b, 2a + 2b, c | 1, 1, 0 |
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- Series of maximal isomorphic subgroups
[p] c' = pc
| R3m (160) | <2; 4> | -b, a + b, pc | | | | p > 1; p ≡ 2 (mod 3)
no conjugate subgroups |
| R3m (160) | <2; 4> | a, b, pc | | | | p > 6; p ≡ 1 (mod 3)
no conjugate subgroups |
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[p2] a' = -pb, b' = pa + pb
| R3m (160) | <2 + (u + v, -u + 2v, 0); 4 + (u + v, u + v, 0)> | -pb, pa + pb, c | u, v, 0 | | | p > 1; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 2 (mod 3) |
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[p2] a' = pa, b' = pb
| R3m (160) | <2 + (u + v, -u + 2v, 0); 4 + (u + v, u + v, 0)> | pa, pb, c | u, v, 0 | | | p > 6; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 1 (mod 3) |
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I Minimal translationengleiche supergroups
| [2] R-3m (166); [4] P-43m (215); [4] F-43m (216); [4] I-43m (217) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [3] a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c P31m (157) |
RHOMBOHEDRAL AXES
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
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| | |
| | (1) x, y, z | (2) z, x, y | (3) y, z, x | | (4) z, y, x | (5) y, x, z | (6) x, z, y |
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I Maximal translationengleiche subgroups
| [2] R31 (146, R3) | 1; 2; 3 |
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| [3] R1m (8, C1m1) | 1; 4 | -a - c, -a + c, a + b + c
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| | [3] R1m (8, C1m1) | 1; 5 | -a - b, a - b, a + b + c
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| | [3] R1m (8, C1m1) | 1; 6 | -b - c, b - c, a + b + c
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II Maximal klassengleiche subgroups
- Loss of centring translations
[2] a' = a + c, b' = a + b, c' = b + c
| R3m (160) | <2; 4> | a + c, a + b, b + c | |
[2] a' = a + b, b' = b + c, c' = a + c
| R3c (161) | <2; 4 + (1, 1, 1)> | a + b, b + c, a + c | |
[3] a' = a - b, b' = b - c, c' = a + b + c
| P3m1 (156) | <2; 4> | a - b, b - c, a + b + c | |
[4] a' = a - b + c, b' = a + b - c, c' = -a + b + c
| R3m (160) | <2; 4> | a - b + c, a + b - c, -a + b + c | | | R3m (160) | <2 + (1, -2, 1); 4 + (1, 0, -1)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R3m (160) | <2 + (1, 1, -2); 4 + (1, 0, -1)> | a - b + c, a + b - c, -a + b + c | 0, 1, -1 | | R3m (160) | <2 + (2, -1, -1); 4 + (2, 0, -2)> | a - b + c, a + b - c, -a + b + c | 1, 0, -1 |
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- Series of maximal isomorphic subgroups
[p] a' = 1/3((p + 1)a + (p - 2)b + (p + 1)c), b' = 1/3((p + 1)a + (p + 1)b + (p - 2)c), c' = 1/3((p - 2)a + (p + 1)b + (p + 1)c)
| R3m (160) | <2; 4> | a' = 1/3((p + 1)a ..., see lattice relations | | | | p > 1; p ≡ 2 (mod 3)
no conjugate subgroups |
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[p] a' = 1/3((p + 2)a + (p - 1)b + (p - 1)c), b' = 1/3((p - 1)a + (p + 2)b + (p - 1)c), c' = 1/3((p - 1)a + (p - 1)b + (p + 2)c)
| R3m (160) | <2; 4> | a' = 1/3((p + 2)a ..., see lattice relations | | | | p > 6; p ≡ 1 (mod 3)
no conjugate subgroups |
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[p2] a' = 1/3((p + 1)a + (1 - 2p)b + (p + 1)c), b' = 1/3((p + 1)a + (p + 1)b + (1 - 2p)c), c' = 1/3((1 - 2p)a + (p + 1)b + (p + 1)c)
| R3m (160) | <2 + (u + v, -2u + v, u - 2v); 4 + (u + v, 0, -u - v)> | a' = 1/3((p + 1)a ..., see lattice relations | u, -u + v, -v | | | p > 1; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 2 (mod 3) |
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[p2] a' = 1/3((2p + 1)a + (1 - p)b + (1 - p)c), b' = 1/3((1 - p)a + (2p + 1)b + (1 - p)c), c' = 1/3((1 - p)a + (1 - p)b + (2p + 1)c)
| R3m (160) | <2 + (u + v, -2u + v, u - 2v); 4 + (u + v, 0, -u - v)> | a' = 1/3((2p + 1)a ..., see lattice relations | u, -u + v, -v | | | p > 6; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 1 (mod 3) |
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I Minimal translationengleiche supergroups
| [2] R-3m (166); [4] P-43m (215); [4] F-43m (216); [4] I-43m (217) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [3] a' = 1/3(2a - b - c), b' = 1/3(a + 2b - c), c' = 1/3(a + b + c) P31m (157) |
