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Axes |
Coordinates |
Wyckoff positions |
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origin 1 |
origin 2 |
2a |
4b |
4c |
6d |
8e |
12f |
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12g |
24h |
| I Maximal translationengleiche subgroups |
| [2] P23 |
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x + (1/4), y + (1/4), z + (1/4) |
1a; 1b |
4e |
4e |
3c; 3d |
2 × 4e |
6f; 6i |
| (195) |
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6g; 6h |
2 × 12j |
| [4] R-3 |
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x - (1/4), y - (1/4), z - (1/4) |
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2c |
1a; 3e |
1b; 3d |
6f |
2c; 6f |
2 × 6f |
| (148) |
(rhombohedral axes) |
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2 × 6f |
4 × 6f |
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a - b, b - c, |
(1/3)(2x - y - z), |
(1/3)(2x - y - z), |
6c |
3a; 9e |
3b; 9d |
18f |
6c; 18f |
2 × 18f |
|
a + b + c |
(1/3)(x + y - 2z), |
(1/3)(x + y - 2z), |
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2 × 18f |
4 × 18f |
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(hexagonal axes) |
(1/3)(x + y + z) - (1/4) |
(1/3)(x + y + z) |
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| conjugate: |
-a - b, b + c, |
(1/3)(-2x - y + z), |
(1/3)(-2x - y + z) + (1/2), |
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-a + b - c |
(1/3)(-x + y + 2z), |
(1/3)( - x + y + 2z) + (1/2), |
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(hexagonal axes) |
(1/3)( - x + y - z) - (1/4) |
(1/3)(-x + y - z) |
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| conjugate: |
a + b, -b + c, |
(1/3)(2x + y + z), |
(1/3)(2x + y + z), |
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a - b - c |
(1/3)(x - y + 2z), |
(1/3)(x - y + 2z) + (1/2), |
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(hexagonal axes) |
(1/3)(x - y - z) - (1/4) |
(1/3)(x - y - z) |
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| conjugate: |
-a + b, -b - c, |
(1/3)(-2x + y - z), |
(1/3)( - 2x + y - z) + (1/2), |
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-a - b + c |
(1/3)(-x - y - 2z), |
(1/3)(-x - y - 2z), |
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(hexagonal axes) |
(1/3)( - x - y + z) - (1/4) |
(1/3)(-x - y + z) |
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| [3] Pnnn |
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2a |
4e(f*) |
4f(e*) |
2b; 2c; 2d |
8m |
4g; 4i; 4k |
| (48) |
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4h; 4j; 4l |
3 × 8m |
| * origin 2 |
| II Maximal klassengleiche subgroups |
| Enlarged unit cell, non-isomorphic |
| [2] Fd-3 |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; |
(1/2)x, (1/2)y, (1/2)z; |
8a; 8b |
16c; 16d |
32e |
48f |
2 × 32e |
2 × 48f |
| (203) |
|
+((1/2), 0, 0) |
+((1/2), 0, 0) |
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96g |
2 × 96g |
| [2] Fd-3 |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), |
(1/2)x + (1/4), (1/2)y + (1/4), |
8a; 8b |
32e |
16c; 16d |
48f |
2 × 32e |
2 × 48f |
| (203) |
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(1/2)z + (1/4); +((1/2), 0, 0) |
(1/2)z + (1/4); +((1/2), 0, 0) |
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96g |
2 × 96g |
| Enlarged unit cell, isomorphic |
| [27] Pn-3 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; |
(1/3)x, (1/3)y, (1/3)z; |
2a; 2 × 8e; |
4b(c† ); 8e; |
4c(b† ); 8e; |
6d; 12f; |
3 × 8e; |
3 × 12f; |
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±((1/3), 0, 0); ±(0, (1/3), 0); ±(0, 0, (1/3)); |
12f; 24h |
4 × 24h |
4 × 24h |
2 × 12g; |
8 × 24h |
12 × 24h |
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±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); ±((1/3), (1/3), 0); |
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5 × 24h |
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±(0, (1/3), (2/3)); ±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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3 × 12g; |
27 × 24h |
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±((1/3), (1/3), (1/3)); ±((1/3), (1/3), (2/3)); ±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) |
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12 × 24h |
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| [p3] Pn-3 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; |
(1/p)x, (1/p)y, (1/p)z; |
2a; |
4b(c‡ ); |
4c(b‡ ); |
6d; ((p - 1)/2) × 12f |
p × 8e; |
p × 12f; |
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+((u/p), (v/p), (w/p)) |
+((u/p), (v/p), (w/p)) |
(p - 1) × 8e; |
((p - 1)/2) × 8e; |
((p - 1)/2) × 8e; |
(p - 1) × 12g; |
((p(p2 - 1))/3) × |
((p(p2 - 1))/2) × |
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p = prime > 2; u, v, w = 1, . . ., p - 1 |
((p - 1)/2) × 12f; |
((p(p2 - 1))/6) × |
((p(p2 - 1))/6) × |
(((p - 1)2(p + 2))/4) × |
24h |
24h |
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((p3 - 7p + 6)/12) × |
24h |
24h |
24h |
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24h |
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p × 12g; |
p3 × 24h |
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((p(p2 - 1))/2) × |
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24h |
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| † origin 1 |
| ‡ origin 1 and p = 4n - 1 |
