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, -y, -x | (5) -y, -x, -z | (6) -x, -z, -y | | (7) -x, -y, -z | (8) -z, -x, -y | (9) -y, -z, -x | | (10) z, y, x | (11) y, x, z | (12) x, z, y |
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I Maximal translationengleiche subgroups
| [2] R3m (160) | 1; 2; 3; 10; 11; 12 |
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| [2] R32 (155) | 1; 2; 3; 4; 5; 6 |
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| [2] R-31 (148, R-3) | 1; 2; 3; 7; 8; 9 |
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 | [3] R12/m (12, C12/m1) | 1; 4; 7; 10 | -a - c, -a + c, a + b + c
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| | [3] R12/m (12, C12/m1) | 1; 5; 7; 11 | -a - b, a - b, a + b + c
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| | [3] R12/m (12, C12/m1) | 1; 6; 7; 12 | -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
| R-3m (166) | <2; 4; 7> | a + c, a + b, b + c | |
| R-3m (166) | <2; (4; 7) + (1, 1, 1)> | a + c, a + b, b + c | 1/2, 1/2, 1/2 |
[2] a' = a + b, b' = b + c, c' = a + c
| R-3c (167) | <2; 7; 4 + (1, 1, 1)> | a + b, b + c, a + c | |
| R-3c (167) | <2; 4; 7 + (1, 1, 1)> | a + b, b + c, a + c | 1/2, 1/2, 1/2 |
[3] a' = a - b, b' = b - c, c' = a + b + c
| P-3m1 (164) | <2; 4; 7> | a - b, b - c, a + b + c | | | P-3m1 (164) | <2 + (1, -1, 0); 4 + (1, 0, 1); 7 + (2, 0, 0)> | a - b, b - c, a + b + c | 1, 0, 0 | | P-3m1 (164) | <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-3m (166) | <2; 4; 7> | a - b + c, a + b - c, -a + b + c | | | R-3m (166) | <(2; 4) + (1, -2, 1); 7 + (2, -2, 0)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R-3m (166) | <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-3m (166) | <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-3m (166) | <2; (4; 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 ≡ 2 (mod 3) |
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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-3m (166) | <2; (4; 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 3) |
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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-3m (166) | <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-3m (166) | <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; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 1 (mod 3) |
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I Minimal translationengleiche supergroups
| [4] Pm-3m (221); [4] Pn-3m (224); [4] Fm-3m (225); [4] Fd-3m (227); [4] Im-3m (229) |
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-31m (162) |
