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] R12 (5, C121) | 1; 4 | -a - c, -a + c, a + b + c
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| | [3] R12 (5, C121) | 1; 5 | -a - b, a - b, a + b + c
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| | [3] R12 (5, C121) | 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
| R32 (155) | <2; 4> | a + c, a + b, b + c | |
| R32 (155) | <2; 4 + (1, 1, 1)> | a + c, a + b, b + c | 1/2, 1/2, 1/2 |
[3] a' = a - b, b' = b - c, c' = a + b + c
| P3221 (154) | <4; 2 + (2, 0, 0)> | a - b, b - c, a + b + c | 2/3, 0, -2/3 | | P3221 (154) | <(2; 4) + (1, 0, 1)> | a - b, b - c, a + b + c | 2/3, 0, 1/3 | | P3221 (154) | <2 + (1, 1, 0); 4 + (1, 2, 1)> | a - b, b - c, a + b + c | 2/3, 1, 1/3 |
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| P3121 (152) | <4; 2 + (1, 0, 0)> | a - b, b - c, a + b + c | 1/3, 0, -1/3 | | P3121 (152) | <2 + (1, -1, 1); 4 + (2, 0, 2)> | a - b, b - c, a + b + c | 4/3, 0, 2/3 | | P3121 (152) | <2 + (1, 1, -1); 4 + (0, 2, 0)> | a - b, b - c, a + b + c | 1/3, 1, -1/3 |
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| P321 (150) | <2; 4> | a - b, b - c, a + b + c | | | P321 (150) | <2 + (1, -1, 0); 4 + (1, 0, 1)> | a - b, b - c, a + b + c | 1, 0, 0 | | P321 (150) | <2 + (1, 0, -1); 4 + (1, 2, 1)> | 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
 | R32 (155) | <2; 4> | a - b + c, a + b - c, -a + b + c | | | R32 (155) | <(2; 4) + (1, -2, 1)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R32 (155) | <2 + (1, 1, -2); 4 + (-1, 2, -1)> | a - b + c, a + b - c, -a + b + c | 0, 1, -1 | | R32 (155) | <4; 2 + (2, -1, -1)> | 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)
| R32 (155) | <2; 4 + (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)
| R32 (155) | <2; 4 + (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)
| R32 (155) | <2 + (u + v, -2u + v, u - 2v); 4 + (u - v, -2u + 2v, 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)
| R32 (155) | <2 + (u + v, -2u + v, u - 2v); 4 + (u - v, -2u + 2v, 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); [2] R-3c (167); [4] P432 (207); [4] P4232 (208); [4] F432 (209); [4] F4132 (210); [4] I432 (211); [4] P4332 (212); [4] P4132 (213); [4] I4132 (214) |
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) P312 (149) |
