|
Axes |
Coordinates |
Wyckoff positions |
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4a |
4b |
8c |
24d |
24e |
32f |
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48g |
48h |
48i |
96j |
| I Maximal translationengleiche subgroups |
| [2] F23 |
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4a |
4b |
4c; 4d |
24g |
24f |
2 × 16e |
| (196) |
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48h |
48h |
2 × 24g |
2 × 48h |
| [4] R32 |
(1/2)(b + c), (1/2)(a + c), |
-x + y + z, x - y + z, |
1a |
1b |
2c |
3d; 3e |
6f |
2c; 6f |
| (155) |
(1/2)(a + b) |
x + y - z |
|
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2 × 3d; 6f |
2 × 3e; 6f |
2 × 6f |
4 × 6f |
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(rhombohedral axes) |
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(1/2)(-a + b), (1/2)(-b + c), |
(2/3)(-2x + y + z), |
3a |
3b |
6c |
9d; 9e |
18f |
6c; 18f |
|
a + b + c |
(2/3)(-x - y + 2z), |
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2 × 9d; 18f |
2 × 9e; 18f |
2 × 18f |
4 × 18f |
|
(hexagonal axes) |
(1/3)(x + y + z) |
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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) |
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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) |
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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) |
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| [3] I422 |
(1/2)(a - b), (1/2)(a + b), c |
x - y, x + y, z |
2a |
2b |
4d |
4c; 8j |
4e; 8g |
16k |
| (97) |
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8h; 16k |
8i; 16k |
8f; 2 × 8j |
3 × 16k |
| conjugate: |
(1/2)(b - c), (1/2)(b + c), a |
y - z, y + z, x |
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| conjugate: |
(1/2)(-a + c), |
-x + z, x + z, y |
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(1/2)(a + c), b |
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| II Maximal klassengleiche subgroups |
| Loss of centring translations |
| [4] P4232 (208) |
x + (1/4), y + (1/4), z + (1/4) |
4b |
4c |
2a; 6d |
6e; 6f; 12h |
24m |
8g; 24m |
| 4 conjugate subgroups |
|
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2 × 12k; 24m |
2 × 12l; 24m |
2 × 12h; 12i; 12j |
4 × 24m |
| [4] P432 (207) |
|
1a; 3c |
1b; 3d |
8g |
12i; 12j |
6e; 6f; 12h |
8g; 24k |
| 4 conjugate subgroups |
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2 × 12i; 24k |
2 × 12j; 24k |
2 × 24k |
4 × 24k |
| Enlarged unit cell, isomorphic |
| [27] F432 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
4a; 24e; |
4b; 24e; |
8c; 2 × 32f; |
24d; 48g; 48h; |
3 × 24e; 6 × 96j |
3 × 32f; |
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±(0, (1/3), 0); ±(0, 0, (1/3)); |
32f; 48g |
32f; 48h |
48i; 96j |
48i; 5 × 96j |
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8 × 96j |
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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; |
3 × 48h; |
3 × 48i; |
27 × 96j |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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12 × 96j |
12 × 96j |
12 × 96j |
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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] F432 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; |
4a; |
4b; |
8c; |
24d; |
p × 24e; |
p × 32f; |
|
p = prime > 2; |
+((u/p), (v/p), (w/p)) |
((p - 1)/2) × 24e; |
((p - 1)/2) × 24e; |
(p - 1) × 32f; |
((p - 1)/2) × 48g; |
((p(p2 - 1))/4) × 96j |
((p(p2 - 1))/3) × |
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u, v, w = 1, . . ., p - 1 |
((p - 1)/2) × 32f; |
((p - 1)/2) × 32f; |
((p - 1)/2) × 48i; |
((p - 1)/2) × 48h; |
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96j |
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((p - 1)/2) × 48g; |
((p - 1)/2) × 48h; |
((p3 - 7p + 6)/12) × |
((p - 1)/2) × 48i; |
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((p3 - 13p + 12)/24) × |
((p3 - 13p + 12)/24) × |
96j |
(((p - 1)2(p + 2))/4) × |
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96j |
96j |
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96j |
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p × 48g; |
p × 48h; |
p × 48i; |
p3 × 48h |
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((p(p2 - 1))/2) × 96j |
((p(p2 - 1))/2) × 96j |
((p(p2 - 1))/2) × 96j |
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