|
Axes |
Coordinates |
Wyckoff positions |
|
|
|
4a |
4b |
4c |
4d |
16e |
24f |
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|
24g |
48h |
96i |
| I Maximal translationengleiche subgroups |
| [2] F23 |
|
|
4a |
4b |
4c |
4d |
16e |
24f |
| (196) |
|
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|
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|
24g |
48h |
2 × 48h |
| [4] R3m |
(1/2)(b + c), (1/2)(a + c), |
-x + y + z, x - y + z, |
1a |
1a |
1a |
1a |
1a; 3b |
2 × 3b |
| (160) |
(1/2)(a + b) |
x + y - z |
|
|
|
2 × 3b |
2 × 3b; 6c |
4 × 6c |
|
(rhombohedral axes) |
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(1/2)(-a + b), (1/2)(-b + c), |
(2/3)(-2x + y + z), |
3a |
3a |
3a |
3a |
3a; 9b |
2 × 9b |
|
a + b + c |
(2/3)(-x - y + 2z), |
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2 × 9b |
2 × 9b; 18c |
4 × 18c |
|
(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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|
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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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|
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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] I-4m2 |
(1/2)(a - b), (1/2)(a + b), c |
x - y, x + y, z |
2a |
2b |
2c |
2d |
8i |
4e; 8g |
| (119) |
|
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|
|
|
4f; 8h |
8i; 16j |
3 × 16j |
| conjugate: |
(1/2)(b - c), (1/2)(b + c), a |
y - z, y + z, x |
|
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|
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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] P-43m |
|
|
1a; 3c |
1b; 3d |
4e |
4e |
4e; 12i |
6f; 6g; 12h |
| (215) |
4 conjugate subgroups |
|
|
|
2 × 12i |
2 × 12i; 24j |
4 × 24j |
| [4] P-43m |
|
x + (1/4), y + (1/4), z + (1/4) |
4e |
4e |
1b; 3d |
1a; 3c |
4e; 12i |
2 × 12i |
| (215) |
4 conjugate subgroups |
|
|
|
6f; 6g; 12h |
2 × 12i; 24j |
4 × 24j |
| Enlarged unit cell, isomorphic |
| [27] F-43m |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
4a; 2 × 16e; |
4b; 2 × 16e; |
4d; 2 × 16e; |
4c; 2 × 16e; |
3 × 16e; |
3 × 24f; |
|
|
±(0, (1/3), 0); ±(0, 0, (1/3)); |
24f; 48h |
24f; 48h |
24g; 48h |
24g; 48h |
6 × 48h; 96i |
6 × 48h; |
|
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±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
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3 × 96i |
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±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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3 × 24g; |
9 × 48h; |
27 × 96i |
|
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±((1/3), (1/3), (1/3)); ±((2/3), (1/3), (1/3)); |
|
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6 × 48h; |
9 × 96i |
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±((1/3), (2/3), (1/3)); ±((1/3), (1/3), (2/3)) |
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3 × 96i |
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| [p3] F-43m |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
4a; |
4b; |
4c(d*); |
4d(c*); |
p × 16e; |
p × 24f; |
|
p = prime > 2; |
u, v, w = 1, . . ., p - 1 |
(p - 1) × 16e; |
(p - 1) × 16e; |
(p - 1) × 16e; |
(p - 1) × 16e; |
p(p - 1) × |
p(p - 1) × |
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|
|
((p - 1)/2) × 24f; |
((p - 1)/2) × 24f; |
((p - 1)/2) × 24g; |
((p - 1)/2) × 24g; |
48h; |
48h; |
|
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|
(((p - 1)(p - 2))/2) × |
(((p - 1)(p - 2))/2) × |
(((p - 1)(p - 2))/2) × |
(((p - 1)(p - 2))/2) × |
((p(p - 1)(p - 2))/6) |
((p(p - 1)2)/4) × |
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48h; |
48h; |
48h; |
48h; |
× 96i |
96i |
|
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|
(((p - 1)(p - 2)(p - 3))/24) |
(((p - 1)(p - 2)(p - 3))/24) |
(((p - 1)(p - 2)(p - 3))/24) |
(((p - 1)(p - 2)(p - 3))/24) |
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× 96i |
× 96i |
× 96i |
× 96i |
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p × 24g; |
p2 × 48h; |
p3 × 96i |
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p(p - 1) × 48h; |
((p2(p - 1))/2) × 96i |
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((p(p - 1)2)/4) × 96i |
|
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| * p = 4n - 1 |
