Origin at centre (-3)
| Asymmetric unit | 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 2/3; 0 ≤ z ≤ 1/6; x ≤ (1 + y)/2; y ≤ min(1 - x, (1 + x)/2) |
| Vertices | | 0, 0, 0 | 1/2, 0, 0 | 2/3, 1/3, 0 | 1/3, 2/3, 0 | 0, 1/2, 0 | | 0, 0, 1/6 | 1/2, 0, 1/6 | 2/3, 1/3, 1/6 | 1/3, 2/3, 1/6 | 0, 1/2, 1/6 |
|
For (0, 0, 0)+ set
| (1) 1 | (2) 3+ 0, 0, z | (3) 3- 0, 0, z |
| (4) -1 0, 0, 0 | (5) -3+ 0, 0, z; 0, 0, 0 | (6) -3- 0, 0, z; 0, 0, 0 |
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 |
| (4) -1 1/3, 1/6, 1/6 | (5) -3+ 1/3, -1/3, z; 1/3, -1/3, 1/6 | (6) -3- 1/3, 2/3, z; 1/3, 2/3, 1/6 |
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 |
| (4) -1 1/6, 1/3, 1/3 | (5) -3+ 2/3, 1/3, z; 2/3, 1/3, 1/3 | (6) -3- -1/3, 1/3, z; -1/3, 1/3, 1/3 |
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 | Reflection conditions |
| | (0, 0, 0)+ (2/3, 1/3, 1/3)+ (1/3, 2/3, 2/3)+ | General:
|
| | (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 |
| 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
|
| | 1/2, 0, 0 | 0, 1/2, 0 | 1/2, 1/2, 0 |
| no extra conditions |
| | 1/2, 0, 1/2 | 0, 1/2, 1/2 | 1/2, 1/2, 1/2 |
| no extra conditions |
| | no extra conditions |
| | no extra conditions |
| | no extra conditions |
Symmetry of special projections
Along [001] p6
a' = 1/3(2a + b) b' = 1/3(-a + b)
Origin at 0, 0, z | Along [100] p2
a' = 1/2(a + 2b) b' = 1/3(-a - 2b + c)
Origin at x, 0, 0 | Along [210] p2
a' = 1/2b b' = 1/3c
Origin at x, 1/2x, 0 |
Maximal non-isomorphic subgroups
| I | | [2] R3 (146) | (1; 2; 3)+ |
| | | [3] R-1 (P-1, 2) | (1; 4)+ |
| IIa |  | [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) | | | [3] P-3 (147) | 1; 2; 3; (4; 5; 6) + (2/3, 1/3, 1/3) |
|
Maximal isomorphic subgroups of lowest index
| IIc | [2] R-3 (a' = -a, b' = -b, c' = 2c) (148); [4] R-3 (a' = -2a, b' = -2b) (148) |
Minimal non-isomorphic supergroups
| I | [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 | [3] P-3 (a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c) (147) |
Origin at centre (-3)
| Asymmetric unit | 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1/2; z ≤ min(x, y, 1 - x, 1 - y) |
| Vertices | | 0, 0, 0 | 1, 0, 0 | 1, 1, 0 | 0, 1, 0 | 1/2, 1/2, 1/2 |
|
| (1) 1 | (2) 3+ x, x, x | (3) 3- x, x, x |
| (4) -1 0, 0, 0 | (5) -3+ x, x, x; 0, 0, 0 | (6) -3- x, x, x; 0, 0, 0 |
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates | Reflection conditions |
| | | General:
|
| | (1) x, y, z | (2) z, x, y | (3) y, z, x | | (4) -x, -y, -z | (5) -z, -x, -y | (6) -y, -z, -x |
| no conditions |
| | | Special: as above, plus
|
| | 0, 1/2, 1/2 | 1/2, 0, 1/2 | 1/2, 1/2, 0 |
| no extra conditions |
| | 1/2, 0, 0 | 0, 1/2, 0 | 0, 0, 1/2 |
| no extra conditions |
| | no extra conditions |
| | no extra conditions |
| | no extra conditions |
Symmetry of special projections
Along [111] p6
a' = 1/3(2a - b - c) b' = 1/3(-a + 2b - c)
Origin at x, x, x | Along [1-10] p2
a' = 1/2(a + b - 2c) b' = c
Origin at x, -x, 0 | Along [2-1-1] p2
a' = 1/2(b - c) b' = 1/3(a + b + c)
Origin at 2x, -x, -x |
Maximal non-isomorphic subgroups
| I | | [2] R3 (146) | 1; 2; 3 |
| | | [3] R-1 (P-1, 2) | 1; 4 |
| IIb | [3] P-3 (a' = a - b, b' = b - c, c' = a + b + c) (147) |
Maximal isomorphic subgroups of lowest index
| IIc | [2] R-3 (a' = b + c, b' = a + c, c' = a + b) (148); [4] R-3 (a' = -a + b + c, b' = a - b + c, c' = a + b - c) (148) |
Minimal non-isomorphic supergroups
| I | [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 | [3] P-3 (a' = 1/3(2a - b - c), b' = 1/3(-a + 2b - c), c' = 1/3(a + b + c)) (147) |
