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) -x, -y, -z | (5) y, -x + y, -z | (6) x - y, x, -z |
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I Maximal translationengleiche subgroups
| [2] R3 (146) | (1; 2; 3)+ |
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| [3] R-1 (2, P-1) | (1; 4)+ | a, b, 1/3(-a - 2b + c)
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II Maximal klassengleiche subgroups
- Loss of centring translations
 | [3] P-3 (147) | 1; 2; 3; 4; 5; 6 | | | | [3] P-3 (147) | 1; 2; 3; (4; 5; 6) + (1/3, 2/3, 2/3) | | 2/3, 1/3, 1/3 | | [3] P-3 (147) | 1; 2; 3; (4; 5; 6) + (2/3, 1/3, 1/3) | | 1/3, 2/3, 2/3 |
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[2] a' = -b, b' = a + b, c' = 2c
| R-3 (148) | <2; 4> | -b, a + b, 2c | |
| R-3 (148) | <2; 4 + (0, 0, 1)> | -b, a + b, 2c | 0, 0, 1/2 |
[4] a' = -2b, b' = 2a + 2b
 | R-3 (148) | <2; 4> | -2b, 2a + 2b, c | | | R-3 (148) | <2 + (1, -1, 0); 4 + (2, 0, 0)> | -2b, 2a + 2b, c | 1, 0, 0 | | R-3 (148) | <2 + (1, 2, 0); 4 + (0, 2, 0)> | -2b, 2a + 2b, c | 0, 1, 0 | | R-3 (148) | <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
| R-3 (148) | <2; 4 + (0, 0, 2u)> | -b, a + b, pc | 0, 0, u | | | p > 4; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 2 (mod 3) |
| R-3 (148) | <2; 4 + (0, 0, 2u)> | a, b, pc | 0, 0, u | | | p > 6; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 1 (mod 3) |
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[p2] a' = -pb, b' = pa + pb
| R-3 (148) | <2 + (u + v, -u + 2v, 0); 4 + (2u, 2v, 0)> | -pb, pa + pb, c | u, v, 0 | | | p > 1; p ≡ 2 (mod 3); 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for the prime p |
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[p = q2 + r2 - qr] a' = (q - r)a - rb, b' = ra + qb
| R-3 (148) | <2 + (u, -u, 0); 4 + (2u, 0, 0)> | (q - r)a - rb, ra + qb, c | u, 0, 0 | | | q > 0; r > 0; q ≠ r; q + r ≡ 1 (mod 3); p > 6; 0 ≤ u < p
p conjugate subgroups for each pair of q and r |
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I Minimal translationengleiche supergroups
| [2] R-3m (166); [2] R-3c (167); [4] Pm-3 (200); [4] Pn-3 (201); [4] Fm-3 (202); [4] Fd-3 (203); [4] Im-3 (204); [4] Pa-3 (205); [4] Ia-3 (206) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [3] a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c P-3 (147) |
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) -x, -y, -z | (5) -z, -x, -y | (6) -y, -z, -x |
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I Maximal translationengleiche subgroups
| [2] R3 (146) | 1; 2; 3 |
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| [3] R-1 (2, P-1) | 1; 4 |
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II Maximal klassengleiche subgroups
- Loss of centring translations
[2] a' = a + c, b' = a + b, c' = b + c
| R-3 (148) | <2; 4> | a + c, a + b, b + c | |
| R-3 (148) | <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
| P-3 (147) | <2; 4> | a - b, b - c, a + b + c | | | P-3 (147) | <2 + (1, -1, 0); 4 + (2, 0, 0)> | a - b, b - c, a + b + c | 1, 0, 0 | | P-3 (147) | <2 + (1, 0, -1); 4 + (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-3 (148) | <2; 4> | a - b + c, a + b - c, -a + b + c | | | R-3 (148) | <2 + (1, -2, 1); 4 + (2, -2, 0)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R-3 (148) | <2 + (1, 1, -2); 4 + (0, 2, -2)> | a - b + c, a + b - c, -a + b + c | 0, 1, -1 | | R-3 (148) | <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)
| R-3 (148) | <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)
| R-3 (148) | <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)
| R-3 (148) | <2 + (u + v, -2u + v, u - 2v); 4 + (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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[p = q2 + r2 - qr] a' = 1/3(αa + βb + γc), b' = 1/3(γa + αb + βc), c' = 1/3(βa + γb + αc); α = 2q - r + 1, β = 1 - q - r, γ = 2r + 1 - q
| R-3 (148) | <2 + (u, -2u, u); 4 + (2u, -2u, 0)> | a' = 1/3(αa + βb + ..., see lattice relations | u, -u, 0 | | | q > 0; r > 0; q ≠ r; q + r ≡ 1 (mod 3); p > 6; 0 ≤ u < p
p conjugate subgroups for each pair of q and r |
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I Minimal translationengleiche supergroups
| [2] R-3m (166); [2] R-3c (167); [4] Pm-3 (200); [4] Pn-3 (201); [4] Fm-3 (202); [4] Fd-3 (203); [4] Im-3 (204); [4] Pa-3 (205); [4] Ia-3 (206) |
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-3 (147) |
