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 + 1/2 | (5) -x + y, y, z + 1/2 | (6) x, x - y, z + 1/2 |
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I Maximal translationengleiche subgroups
| [2] R31 (146, R3) | (1; 2; 3)+ |
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 | [3] R1c (9, C1c1) | (1; 4)+ | 1/3(-a + b - 2c), -a - b, c
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| | [3] R1c (9, C1c1) | (1; 5)+ | 1/3(-a - 2b - 2c), a, c
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| | [3] R1c (9, C1c1) | (1; 6)+ | 1/3(2a + b - 2c), b, c
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II Maximal klassengleiche subgroups
- Loss of centring translations
| [3] P3c1 (158) | 1; 2; 3; 4; 5; 6 | | |
[4] a' = -2b, b' = 2a + 2b
 | R3c (161) | <2; 4> | -2b, 2a + 2b, c | | | R3c (161) | <2 + (1, -1, 0); 4 + (1, 1, 0)> | -2b, 2a + 2b, c | 1, 0, 0 | | R3c (161) | <2 + (1, 2, 0); 4 + (1, 1, 0)> | -2b, 2a + 2b, c | 0, 1, 0 | | R3c (161) | <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
| R3c (161) | <2; 4 + (0, 0, p/2 - 1/2)> | -b, a + b, pc | | | | p > 4; p ≡ 5 (mod 6)
no conjugate subgroups |
| R3c (161) | <2; 4 + (0, 0, p/2 - 1/2)> | a, b, pc | | | | p > 6; p ≡ 1 (mod 6)
no conjugate subgroups |
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[p2] a' = -pb, b' = pa + pb
| R3c (161) | <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
| R3c (161) | <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-3c (167); [4] P-43n (218); [4] F-43c (219); [4] I-43d (220) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [3] a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c P31c (159); [2] a' = -a, b' = -b, c' = 1/2c R3m (160) |
