|
Axes |
Coordinates |
Wyckoff positions |
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4a |
4b |
4c |
4d |
16e |
24f |
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24g |
48h |
| I Maximal translationengleiche subgroups |
| [4] R3 |
(1/2)(b+c), (1/2)(a+c), |
-x + y + z, |
1a |
1a |
1a |
1a |
1a; 3b |
2 × 3b |
| (146) |
(1/2)(a+b) |
x - y + z, |
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2 × 3b |
4 × 3b |
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(rhombohedral axes) |
x + y - z |
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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 |
4 × 9b |
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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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| 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] F222 |
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4a |
4b |
4c |
4d |
16k |
8e; 8f; 8g |
| (22) |
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8h; 8i; 8j |
3 × 16k |
| II Maximal klassengleiche subgroups |
| Loss of centring translations |
| [4] P213 (198) |
|
4a |
4a |
4a |
4a |
4a; 12b |
2 × 12b |
| 4 conjugate subgroups |
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2 × 12b |
4 × 12b |
| [4] P213 (198) |
x + (1/4), y + (1/4), z + (1/4) |
4a |
4a |
4a |
4a |
4a; 12b |
2 × 12b |
| 4 conjugate subgroups |
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2 × 12b |
4 × 12b |
| [4] P23 (195) |
|
1a; 3c |
1b; 3d |
4e |
4e |
4e; 12j |
6f; 6g; 6h; 6i |
| 4 conjugate subgroups |
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|
2 × 12j |
4 × 12j |
| [4] P23 (195) |
x + (1/4), y + (1/4), z + (1/4) |
4e |
4e |
1b; 3d |
1a; 3c |
4e; 12j |
2 × 12j |
| 4 conjugate subgroups |
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6f; 6g; 6h; 6i |
4 × 12j |
| Enlarged unit cell, isomorphic |
| [27] F23 |
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; |
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±(0, (1/3), 0); ±(0, 0, (1/3)); |
24f; 48h |
24f; 48h |
24g; 48h |
24g; 48h |
8 × 48h |
12 × 48h |
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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 × 24g; |
27 × 48h |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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12 × 48h |
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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] F23 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; |
4a; |
4b; |
4c(d*); |
4d(c*); |
p × 16e; |
p × 24f; |
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+((u/p), (v/p), (w/p)) |
(p - 1) × 16e; |
(p - 1) × 16e; |
(p - 1) × 16e; |
(p - 1) × 16e; |
((p(p2 - 1))/3) × 48h |
((p(p2 - 1))/2) × 48h |
|
p = prime > 2; u, v, w = 1, . . ., p - 1 |
((p - 1)/2) × 24f; |
((p - 1)/2) × 24f; |
((p - 1)/2) × 24g; |
((p - 1)/2) × 24g; |
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((p3 - 7p + 6)/12) × |
((p3 - 7p + 6)/12) × |
((p3 - 7p + 6)/12) × |
((p3 - 7p + 6)/12) × |
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48h |
48h |
48h |
48h |
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p × 24g; |
p3 × 48h |
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((p(p2 - 1))/2) × 48h |
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| * p = 4n - 1 |
