|
Axes |
Coordinates |
Wyckoff positions |
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8a |
8b |
16c |
16d |
32e |
48f |
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48g |
96h |
| I Maximal translationengleiche subgroups |
| [2] F23 |
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4a; 4c |
4b; 4d |
16e |
16e |
2 × 16e |
24f; 24g |
| (196) |
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48h |
2 × 48h |
| [4] R32 |
(1/2)(b + c), (1/2)(a + c), |
-x + y + z - (1/8), |
2c |
2c |
1a; 3e |
1b; 3d |
2c; 6f |
2 × 6f |
| (155) |
(1/2)(a + b) |
x - y + z - (1/8), |
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3d; 3e; 6f |
4 × 6f |
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(rhombohedral axes) |
x + y - z - (1/8) |
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(1/2)(-a + b), (1/2)(-a + c), |
(2/3)(-2x + y + z), |
6c |
6c |
3a; 9e |
3b; 9d |
6c; 18f |
2 × 18f |
|
a + b + c |
(2/3)(-x - y + 2z), |
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9d; 9e; 18f |
4 × 18f |
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(hexagonal axes) |
(1/3)(x + y + z) - (1/8) |
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| conjugate: |
(1/2)(a + b), (1/2)(-b - c), |
(2/3)(2x + y - z), |
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-a + b - c |
(2/3)(x - y - 2z), |
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(hexagonal axes) |
(1/3)(-x + y - z) - (1/8) |
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| conjugate: |
(1/2)(-a - b), (1/2)(b - c), |
(2/3)(-2x - y - z), |
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a - b - c |
(2/3)(-x + y - 2z), |
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(hexagonal axes) |
(1/3)(x - y - z) - (1/8) |
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| conjugate: |
(1/2)(a - b), (1/2)(b + c), |
(2/3)(2x - y + z), |
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-a - b + c |
(2/3)(x + y + 2z), |
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(hexagonal axes) |
(1/3)(-x - y + z) - (1/8) |
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| [3] I4122 |
(1/2)(a - b), (1/2)(a + b), c |
x - y, x + y, z |
4a |
4b |
8f |
8f |
16g |
8c; 8d; 8e |
| (98) |
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8f; 16g |
3 × 16g |
| conjugate: |
(1/2)(b - c), (1/2)(b + c), a |
y - z, y + z, x |
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| conjugate: |
(1/2)(-a + c), (1/2)(a + c), b |
-x + z, x + z, y |
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| II Maximal klassengleiche subgroups |
| Loss of centring translations |
| [4] P4132 |
|
x + (1/4), y + (1/4), z + (1/4) |
8c |
8c |
4a; 12d |
4b; 12d |
8c; 24e |
2 × 24e |
| (213) |
4 conjugate subgroups |
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2 × 12d; 24e |
4 × 24e |
| [4] P4332 |
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8c |
8c |
4a; 12d |
4b; 12d |
8c; 24e |
2 × 24e |
| (212) |
4 conjugate subgroups |
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2 × 12d; 24e |
4 × 24e |
| Enlarged unit cell, isomorphic |
| [27] F4132 |
3b, -3a, 3c |
(1/3)y, -(1/3)x, (1/3)z; |
8a; 2 × 32e; |
8b; 2 × 32e; |
16d; 32e; |
16c; 32e; |
3 × 32e; |
3 × 48f; |
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±((1/3), 0, 0); |
48f; 96h |
48f; 96h |
2 × 48g; |
2 × 48g; |
8 × 96h |
12 × 96h |
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±(0, (1/3), 0); ±(0, 0, (1/3)); |
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3 × 96h |
3 × 96h |
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±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
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±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
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3 × 48g; |
27 × 96h |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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12 × 96h |
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±((1/3), (1/3), (1/3)); ±((1/3), (1/3), (2/3)); |
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±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) |
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| [p3] F4132 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; |
8a; |
8b; |
16c(d*); |
16d(c*); |
p × 32e; |
p × 48f; |
|
p = prime = 4n + 1 |
+((u/p), (v/p), (w/p)) |
(p - 1) × 32e; |
(p - 1) × 32e; |
((p - 1)/2) × 32e; |
((p - 1)/2) × 32e; |
((p(p2 - 1))/3) × 96h |
((p(p2 - 1))/2) × 96h |
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((p - 1)/2) × 48f; |
((p - 1)/2) × 48f; |
(p - 1) × 48g; |
(p - 1) × 48g; |
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pb, -pa, pc |
(1/p)y, -(1/p)x, (1/p)z; |
((p3 - 7p + 6)/12) × |
((p3 - 7p + 6)/12) × |
((p3 - 4p + 3)/6) × |
((p3 - 4p + 3)/6) × |
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p = prime = 4n - 1 |
+((u/p), (v/p), (w/p)) |
96h |
96h |
96h |
96h |
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u, v, w = 1, . . ., p - 1 |
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p × 48g; |
p3 × 96h |
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((p(p2 - 1))/2) × 96h |
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| * p = 8n± 3 |
