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); (7)
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 | | (7) -x, -y, -z | (8) y, -x + y, -z | (9) x - y, x, -z | | (10) -y, -x, z + 1/2 | (11) -x + y, y, z + 1/2 | (12) x, x - y, z + 1/2 |
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I Maximal translationengleiche subgroups
| [2] R3c (161) | (1; 2; 3; 10; 11; 12)+ |
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| [2] R32 (155) | (1; 2; 3; 4; 5; 6)+ |
| 0, 0, 1/4
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| [2] R-31 (148, R-3) | (1; 2; 3; 7; 8; 9)+ |
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 | [3] R12/c (15, C12/c1) | (1; 4; 7; 10)+ | 1/3(-a + b - 2c), -a - b, c
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| | [3] R12/c (15, C12/c1) | (1; 5; 7; 11)+ | 1/3(-a - 2b - 2c), a, c
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| | [3] R12/c (15, C12/c1) | (1; 6; 7; 12)+ | 1/3(2a + b - 2c), b, c
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II Maximal klassengleiche subgroups
- Loss of centring translations
| [3] P-3c1 (165) | 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12 | | | | [3] P-3c1 (165) | 1; 2; 3; 10; 11; 12; (4; 5; 6; 7; 8; 9) + (2/3, 1/3, 1/3) | | 1/3, 2/3, 2/3 | | [3] P-3c1 (165) | 1; 2; 3; 10; 11; 12; (4; 5; 6; 7; 8; 9) + (1/3, 2/3, 2/3) | | 2/3, 1/3, 1/3 |
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[4] a' = -2b, b' = 2a + 2b
 | R-3c (167) | <2; 4; 7> | -2b, 2a + 2b, c | | | R-3c (167) | <(2; 4) + (1, -1, 0); 7 + (2, 0, 0)> | -2b, 2a + 2b, c | 1, 0, 0 | | R-3c (167) | <2 + (1, 2, 0); 4 + (-1, 1, 0); 7 + (0, 2, 0)> | -2b, 2a + 2b, c | 0, 1, 0 | | R-3c (167) | <4; 2 + (2, 1, 0); 7 + (2, 2, 0)> | -2b, 2a + 2b, c | 1, 1, 0 |
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- Series of maximal isomorphic subgroups
[p] c' = pc
| R-3c (167) | <2; 4 + (0, 0, p/2 - 1/2 + 2u); 7 + (0, 0, 2u)> | -b, a + b, pc | 0, 0, u | | | p > 4; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 5 (mod 6) |
| R-3c (167) | <2; 4 + (0, 0, p/2 - 1/2 + 2u); 7 + (0, 0, 2u)> | a, b, pc | 0, 0, u | | | p > 6; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 1 (mod 6) |
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[p2] a' = -pb, b' = pa + pb
| R-3c (167) | <2 + (u + v, -u + 2v, 0); 4 + (u - v, -u + v, 0); 7 + (2u, 2v, 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
| R-3c (167) | <2 + (u + v, -u + 2v, 0); 4 + (u - v, -u + v, 0); 7 + (2u, 2v, 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
| [4] Pn-3n (222); [4] Pm-3n (223); [4] Fm-3c (226); [4] Fd-3c (228); [4] Ia-3d (230) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [3] a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c P-31c (163); [2] a' = -a, b' = -b, c' = 1/2c R-3m (166) |
RHOMBOHEDRAL AXES
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4); (7)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
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| | (1) x, y, z | (2) z, x, y | (3) y, z, x | | (4) -z + 1/2, -y + 1/2, -x + 1/2 | (5) -y + 1/2, -x + 1/2, -z + 1/2 | (6) -x + 1/2, -z + 1/2, -y + 1/2 | | (7) -x, -y, -z | (8) -z, -x, -y | (9) -y, -z, -x | | (10) z + 1/2, y + 1/2, x + 1/2 | (11) y + 1/2, x + 1/2, z + 1/2 | (12) x + 1/2, z + 1/2, y + 1/2 |
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I Maximal translationengleiche subgroups
| [2] R3c (161) | 1; 2; 3; 10; 11; 12 |
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| [2] R32 (155) | 1; 2; 3; 4; 5; 6 |
| 1/4, 1/4, 1/4
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| [2] R-31 (148, R-3) | 1; 2; 3; 7; 8; 9 |
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| [3] R12/c (15, C12/c1) | 1; 4; 7; 10 | -a - c, -a + c, a + b + c
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| | [3] R12/c (15, C12/c1) | 1; 5; 7; 11 | -a - b, a - b, a + b + c
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| | [3] R12/c (15, C12/c1) | 1; 6; 7; 12 | -b - c, b - c, a + b + c
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II Maximal klassengleiche subgroups
- Loss of centring translations
[3] a' = a - b, b' = b - c, c' = a + b + c
| P-3c1 (165) | <2; 4; 7> | a - b, b - c, a + b + c | | | P-3c1 (165) | <2 + (1, -1, 0); 4 + (1, 0, 1); 7 + (2, 0, 0)> | a - b, b - c, a + b + c | 1, 0, 0 | | P-3c1 (165) | <2 + (1, 0, -1); 4 + (1, 2, 1); 7 + (2, 2, 0)> | a - b, b - c, a + b + c | 1, 1, 0 |
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[4] a' = a - b + c, b' = a + b - c, c' = -a + b + c
| R-3c (167) | <2; 4; 7> | a - b + c, a + b - c, -a + b + c | | | R-3c (167) | <(2; 4) + (1, -2, 1); 7 + (2, -2, 0)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R-3c (167) | <2 + (1, 1, -2); 4 + (-1, 2, -1); 7 + (0, 2, -2)> | a - b + c, a + b - c, -a + b + c | 0, 1, -1 | | R-3c (167) | <4; 2 + (2, -1, -1); 7 + (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)
| R-3c (167) | <2; 4 + (p/2 - 1/2 + 2u, p/2 - 1/2 + 2u, p/2 - 1/2 + 2u); 7 + (2u, 2u, 2u)> | a' = 1/3((p + 1)a ..., see lattice relations | u, u, u | | | p > 4; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 5 (mod 6) |
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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)
| R-3c (167) | <2; 4 + (p/2 - 1/2 + 2u, p/2 - 1/2 + 2u, p/2 - 1/2 + 2u); 7 + (2u, 2u, 2u)> | a' = 1/3((p + 2)a ..., see lattice relations | u, u, u | | | p > 6; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 1 (mod 6) |
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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)
| R-3c (167) | <2 + (u + v, -2u + v, u - 2v); 4 + (u - v, -2u + 2v, u - v); 7 + (2u, -2u + 2v, -2v)> | 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)
| R-3c (167) | <2 + (u + v, -2u + v, u - 2v); 4 + (u - v, -2u + 2v, u - v); 7 + (2u, -2u + 2v, -2v)> | a' = 1/3((2p + 1)a ..., see lattice relations | u, -u + v, -v | | | p > 6; p ≡ 1 (mod 3); 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 1 (mod 3) |
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I Minimal translationengleiche supergroups
| [4] Pn-3n (222); [4] Pm-3n (223); [4] Fm-3c (226); [4] Fd-3c (228); [4] Ia-3d (230) |
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) P-31c (163); [2] a' = 1/2(-a + b + c), b' = 1/2(a - b + c), c' = 1/2(a + b - c) R-3m (166) |
