| R3 | C34 | 3 | Trigonal  |
| No. 146 | R3 | Patterson symmetry R-3 | |
| HEXAGONAL AXES | |||
Origin on 3
| Asymmetric unit | 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 2/3; 0 ≤ z ≤ 1/3; x ≤ (1 + y)/2; y ≤ min(1 - x, (1 + x)/2) | ||||||||||
| Vertices |
|
Symmetry operations
For (0, 0, 0)+ set
| (1) 1 | (2) 3+ 0, 0, z | (3) 3- 0, 0, z |
For (2/3, 1/3, 1/3)+ set
| (1) t(2/3, 1/3, 1/3) | (2) 3+(0, 0, 1/3) 1/3, 1/3, z | (3) 3-(0, 0, 1/3) 1/3, 0, z |
For (1/3, 2/3, 2/3)+ set
| (1) t(1/3, 2/3, 2/3) | (2) 3+(0, 0, 2/3) 0, 1/3, z | (3) 3-(0, 0, 2/3) 1/3, 1/3, z |
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(2/3, 1/3, 1/3); (2)
Positions
| Multiplicity, Wyckoff letter, Site symmetry | Coordinates | Reflection conditions | ||||||
| (0, 0, 0)+ (2/3, 1/3, 1/3)+ (1/3, 2/3, 2/3)+ | General: | |||||||
|
| hkil : -h + k + l = 3n hki0 : -h + k = 3n hh(-2h)l : l = 3n h-h0l : h + l = 3n 000l : l = 3n h-h00 : h = 3n |
| Special: as above, plus | ||||||
|
| no extra conditions |
Symmetry of special projections
| Along [001] p3 a' = 1/3(2a + b) b' = 1/3(-a + b) Origin at 0, 0, z | Along [100] p1 a' = 1/2(a + 2b) b' = 1/3(-a - 2b + c) Origin at x, 0, 0 | Along [210] p1 a' = 1/2b b' = 1/3c Origin at x, 1/2x, 0 |
Maximal non-isomorphic subgroups
| I | [3] R1 (P1, 1) | 1+ |
| IIa | [3] P32 (145) | 1; 2 + (1/3, 2/3, 2/3); 3 + (2/3, 1/3, 1/3) | |
| [3] P31 (144) | 1; 2 + (2/3, 1/3, 1/3); 3 + (1/3, 2/3, 2/3) | ||
| [3] P3 (143) | 1; 2; 3 |
| IIb | none |
Maximal isomorphic subgroups of lowest index
| IIc | [2] R3 (a' = -a, b' = -b, c' = 2c) (146); [4] R3 (a' = -2a, b' = -2b) (146) |
Minimal non-isomorphic supergroups
| I | [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 | [3] P3 (a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c) (143) |
| R3 | C34 | 3 | Trigonal |
| No. 146 | R3 | Patterson symmetry R-3 | |
| RHOMBOHEDRAL AXES | |||
Origin on 3
| Asymmetric unit | 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; z ≤ min(x, y) | |||||
| Vertices |
|
Symmetry operations
| (1) 1 | (2) 3+ x, x, x | (3) 3- x, x, x |
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2)
Positions
| Multiplicity, Wyckoff letter, Site symmetry | Coordinates | Reflection conditions | ||||||
| General: | ||||||||
|
| no conditions |
| Special: as above, plus | ||||||
|
| no extra conditions |
Symmetry of special projections
| Along [111] p3 a' = 1/3(2a - b - c) b' = 1/3(-a + 2b - c) Origin at x, x, x | Along [1-10] p1 a' = 1/2(a + b - 2c) b' = c Origin at x, -x, 0 | Along [2-1-1] p1 a' = 1/2(b - c) b' = 1/3(a + b + c) Origin at 2x, -x, -x |
Maximal non-isomorphic subgroups
| I | [3] R1 (P1, 1) | 1 |
| IIa | none |
| IIb | [3] P32 (a' = a - b, b' = b - c, c' = a + b + c) (145); [3] P31 (a' = a - b, b' = b - c, c' = a + b + c) (144); [3] P3 (a' = a - b, b' = b - c, c' = a + b + c) (143) |
Maximal isomorphic subgroups of lowest index
| IIc | [2] R3 (a' = b + c, b' = a + c, c' = a + b) (146); [4] R3 (a' = -a + b + c, b' = a - b + c, c' = a + b - c) (146) |
Minimal non-isomorphic supergroups
| I | [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 | [3] P3 (a' = 1/3(2a - b - c), b' = 1/3(-a + 2b - c), c' = 1/3(a + b + c)) (143) |
