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) |
