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)
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 |
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I Maximal translationengleiche subgroups
| [3] R1 (1, P1) | 1+ | a, b, 1/3(-a - 2b + c)
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II Maximal klassengleiche subgroups
- Loss of centring translations
| [3] P32 (145) | 1; 2 + (1/3, 2/3, 2/3); 3 + (2/3, 1/3, 1/3) | | 0, 1/3, 0 |
| [3] P31 (144) | 1; 2 + (2/3, 1/3, 1/3); 3 + (1/3, 2/3, 2/3) | | 1/3, 1/3, 0 |
| [3] P3 (143) | 1; 2; 3 | | |
[2] a' = -b, b' = a + b, c' = 2c
[4] a' = -2b, b' = 2a + 2b
 | R3 (146) | <2> | -2b, 2a + 2b, c | | | R3 (146) | <2 + (1, -1, 0)> | -2b, 2a + 2b, c | 1, 0, 0 | | R3 (146) | <2 + (1, 2, 0)> | -2b, 2a + 2b, c | 0, 1, 0 | | R3 (146) | <2 + (2, 1, 0)> | -2b, 2a + 2b, c | 1, 1, 0 |
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- Series of maximal isomorphic subgroups
[p] c' = pc
| R3 (146) | <2> | -b, a + b, pc | | | | p > 1; p ≡ 2 (mod 3)
no conjugate subgroups |
| R3 (146) | <2> | a, b, pc | | | | p > 6; p ≡ 1 (mod 3)
no conjugate subgroups |
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[p2] a' = -pb, b' = pa + pb
| R3 (146) | <2 + (u + v, -u + 2v, 0)> | -pb, pa + pb, c | u, v, 0 | | | 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' = (q - r)a - rb, b' = ra + qb
| R3 (146) | <2 + (u, -u, 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-3 (148); [2] R32 (155); [2] R3m (160); [2] R3c (161); [4] P23 (195); [4] F23 (196); [4] I23 (197); [4] P213 (198); [4] I213 (199) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [3] a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c P3 (143) |
RHOMBOHEDRAL AXES
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
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| | |
| | (1) x, y, z | (2) z, x, y | (3) y, z, x |
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I Maximal translationengleiche subgroups
II Maximal klassengleiche subgroups
- Loss of centring translations
[2] a' = a + c, b' = a + b, c' = b + c
| R3 (146) | <2> | a + c, a + b, b + c | |
[3] a' = a - b, b' = b - c, c' = a + b + c
| P32 (145) | <2 + (1, 1, 0)> | a - b, b - c, a + b + c | 0, 1/3, -1/3 |
| P31 (144) | <2 + (1, 0, 0)> | a - b, b - c, a + b + c | 1/3, 0, -1/3 |
| P3 (143) | <2> | a - b, b - c, a + b + c | |
[4] a' = a - b + c, b' = a + b - c, c' = -a + b + c
| R3 (146) | <2> | a - b + c, a + b - c, -a + b + c | | | R3 (146) | <2 + (1, -2, 1)> | a - b + c, a + b - c, -a + b + c | 1, -1, 0 | | R3 (146) | <2 + (1, 1, -2)> | a - b + c, a + b - c, -a + b + c | 0, 1, -1 | | R3 (146) | <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)
| R3 (146) | <2> | a' = 1/3((p + 1)a ..., see lattice relations | | | | p > 1; p ≡ 2 (mod 3)
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)
| R3 (146) | <2> | a' = 1/3((p + 2)a ..., see lattice relations | | | | p > 6; p ≡ 1 (mod 3)
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)
| R3 (146) | <2 + (u + v, -2u + v, u - 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
| R3 (146) | <2 + (u, -2u, u)> | 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-3 (148); [2] R32 (155); [2] R3m (160); [2] R3c (161); [4] P23 (195); [4] F23 (196); [4] I23 (197); [4] P213 (198); [4] I213 (199) |
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) P3 (143) |
