|
Axes |
Coordinates |
Wyckoff positions |
|
|
|
4a |
4b |
8c |
12d |
24e |
| I Maximal translationengleiche subgroups |
| [2] P213 (198) |
|
|
4a |
4a |
2 × 4a |
12b |
2 × 12b |
| [4] R32 (155) |
|
x - (3/8), y - (3/8), z - (3/8) |
1a; 3e |
1b; 3d |
2c; 6f |
3d; 3e; 6f |
4 × 6f |
|
(rhombohedral axes) |
|
|
|
|
|
|
a - b, b - c, |
(1/3)(2x - y - z), |
3a; 9e |
3b; 9d |
6c; 18f |
9d; 9e; 18f |
4 × 18f |
|
a + b + c |
(1/3)(x + y - 2z), |
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(x + y + z) - (3/8) |
|
|
|
|
|
| conjugate: |
-a - b, b + c, |
(1/3)(-2x - y + z), |
|
|
|
|
|
|
-a + b - c |
(1/3)(-x + y + 2z) - (1/2), |
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(-x + y - z) - (3/8) |
|
|
|
|
|
| conjugate: |
a + b, -b + c, |
(1/3)(2x + y + z) - (1/2), |
|
|
|
|
|
|
a - b - c |
(1/3)(x - y + 2z), |
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(x - y - z) - (3/8) |
|
|
|
|
|
| conjugate: |
-a + b, -b - c, |
(1/3)(-2x + y - z) + (1/2), |
|
|
|
|
|
|
-a - b + c |
(1/3)(-x - y - 2z) + (1/2), |
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(-x - y + z) - (3/8) |
|
|
|
|
|
| [3] P41212 (92) |
|
x + (1/4), y, z - (1/8) |
4a |
4a |
8b |
4a; 8b |
3 × 8b |
| conjugate: |
b, c, a |
y + (1/4), z, x - (1/8) |
|
|
|
|
|
| conjugate: |
c, a, b |
z + (1/4), x, y - (1/8) |
|
|
|
|
|
| II Maximal klassengleiche subgroups |
| Enlarged unit cell, isomorphic |
| [27] P4332 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
4a; 8c; 2 × 12d; |
4b; 8c; 2 × 12d; |
3 × 8c; 8 × 24e |
3 × 12d; |
27 × 24e |
| (212) |
|
±(0, (1/3), 0); ±(0, 0, (1/3)); |
3 × 24e |
3 × 24e |
|
12 × 24e |
|
|
|
±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
|
|
|
|
|
|
|
±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
|
|
|
|
|
|
|
±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
|
|
|
|
|
|
|
±((1/3), (1/3), (1/3)); ±((2/3), (1/3), (1/3)); |
|
|
|
|
|
|
|
±((1/3), (2/3), (1/3)); ±((1/3), (1/3), (2/3)) |
|
|
|
|
|
| [p3] P4132 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
4a(b*); ((p - 1)/2) × 8c; |
4b(a*); ((p - 1)/2) × 8c; |
p × 8c; |
p × 12d; |
p3 × 24e |
|
p = prime = 4n + 1; |
(p - 1) × 12d; |
(p - 1) × 12d; |
((p(p2 - 1))/3) × 24e |
((p(p2 - 1))/2) × 24e |
|
|
u, v, w = 1, . . ., p - 1 |
((p3 - 4p + 3)/6) × 24e |
((p3 - 4p + 3)/6) × 24e |
|
|
|
| [p3] P4332 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
4a(b† ); ((p - 1)/2) × 8c; |
4b(a† ); ((p - 1)/2) × 8c; |
p × 8c; |
p × 12d; |
p3 × 24e |
| (212) |
p = prime = 4n - 1; |
(p - 1) × 12d; |
(p - 1) × 12d; |
((p(p2 - 1))/3) × 24e |
((p(p2 - 1))/2) × 24e |
|
|
u, v, w = 1, . . ., p - 1 |
((p3 - 4p + 3)/6) × 24e |
((p3 - 4p + 3)/6) × 24e |
|
|
|
| * p = 8n + 5 |
† p = 8n - 1 |
