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 + 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 |
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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 | -a - c, -a + c, a + b + c
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| | [3] R1c (9, C1c1) | 1; 5 | -a - b, a - b, a + b + c
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| | [3] R1c (9, C1c1) | 1; 6 | -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
| P3c1 (158) | <2; 4> | a - b, b - c, a + b + c | |
[4] a' = a - b + c, b' = a + b - c, c' = -a + b + c
 | R3c (161) | <2; 4> | a - b + c, a + b - c, -a + b + c | | | R3c (161) | <2 + (1, -2, 1); 4 + (1, 0, -1)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R3c (161) | <2 + (1, 1, -2); 4 + (1, 0, -1)> | a - b + c, a + b - c, -a + b + c | 0, 1, -1 | | R3c (161) | <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)
| R3c (161) | <2; 4 + (p/2 - 1/2, p/2 - 1/2, p/2 - 1/2)> | a' = 1/3((p + 1)a ..., see lattice relations | | | | p > 4; p ≡ 5 (mod 6)
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)
| R3c (161) | <2; 4 + (p/2 - 1/2, p/2 - 1/2, p/2 - 1/2)> | a' = 1/3((p + 2)a ..., see lattice relations | | | | p > 6; p ≡ 1 (mod 6)
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)
| R3c (161) | <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)
| R3c (161) | <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-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 - c), b' = 1/3(-a + 2b - c), c' = 1/3(a + b + c) P31c (159); [2] a' = 1/2(-a + b + c), b' = 1/2(a - b + c), c' = 1/2(a + b - c) R3m (160) |
