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Axes |
Coordinates |
Wyckoff positions |
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1a |
1b |
2c |
2d |
3e |
3f |
6g |
| I Maximal translationengleiche subgroups |
| [2] P3 (143) |
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1a |
1a |
2 × 1a |
1b; 1c |
3d |
3d |
2 × 3d |
| [3] P-1 (2) |
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1a |
1b |
2i |
2i |
1c; 1d; 1e |
1f; 1g; 1h |
3 × 2i |
| II Maximal klassengleiche subgroups |
| Enlarged unit cell, non-isomorphic |
| [3] R-3 |
a - b, a + 2b, 3c |
(1/3)(2x - y), (1/3)(x + y), (1/3)z; |
3a; 6c |
3b; 6c |
3 × 6c |
18f |
9e; 18f |
9d; 18f |
3 × 18f |
| (148) |
(hexagonal |
±(0, 0, (1/3)) |
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axes) |
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| [3] R-3 |
2a + b, -a + b, 3c |
(1/3)(x + y), (1/3)(-x + 2y), (1/3)z; |
3a; 6c |
3b; 6c |
3 × 6c |
18f |
9e; 18f |
9d; 18f |
3 × 18f |
| (148) |
(hexagonal |
±(0, 0, (1/3)) |
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axes) |
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| Enlarged unit cell, isomorphic |
| [2] P-3 |
a, b, 2c |
x, y, (1/2)z; +(0, 0, (1/2)) |
1a; 1b |
2c |
2 × 2c |
2 × 2d |
3e; 3f |
6g |
2 × 6g |
| [2] P-3 |
a, b, 2c |
x, y, (1/2)z + (1/4); +(0, 0, (1/2)) |
2c |
1a; 1b |
2 × 2c |
2 × 2d |
6g |
3e; 3f |
2 × 6g |
| [3] P-3 |
a, b, 3c |
x, y, (1/3)z; ±(0, 0, (1/3)) |
1a; 2c |
1b; 2c |
3 × 2c |
3 × 2d |
3e; 6g |
3f; 6g |
3 × 6g |
| [p] P-3 |
a, b, pc |
x, y, (1/p)z; +(0, 0, (u/p)) |
1a; |
1b; |
p × 2c |
p × 2d |
3e; |
3f; |
p × 6g |
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p = prime > 2; u = 1, . . ., p - 1 |
((p - 1)/2) × 2c |
((p - 1)/2) × 2c |
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((p - 1)/2) × 6g |
((p - 1)/2) × 6g |
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| [3] P-3 |
2a + b, -a + b, c |
(1/3)(x + y), (1/3)(-x + 2y), z; |
1a; 2d |
1b; 2d |
2c; 2 × 2d |
6g |
3e; 6g |
3f; 6g |
3 × 6g |
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±((1/3), (2/3), 0) |
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| [7] P-3 |
3a + b, -a + 2b, c |
(1/7)(2x + y), (1/7)(-x + 3y), z; |
1a; 6g |
1b; 6g |
2c; 2 × 6g |
2d; 2 × 6g |
3e; 3 × 6g |
3f; 3 × 6g |
7 × 6g |
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±((1/7), (3/7), 0); ±((3/7), (2/7), 0); |
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±((5/7), (1/7), 0) |
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| [7] P-3 |
3a + 2b, -2a + b, c |
(1/7)(x + 2y), (1/7)(-2x + 3y), z; |
1a; 6g |
1b; 6g |
2c; 2 × 6g |
2d; 2 × 6g |
3e; 3 × 6g |
3f; 3 × 6g |
7 × 6g |
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±((2/7), (3/7), 0); ±((3/7), (1/7), 0); |
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±((1/7), (5/7), 0) |
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| [p] P-3 |
qa + rb, |
(1/p)((q - r)x + ry), |
1a; |
1b; |
2c; |
2d; |
3e; |
3f; |
p × 6g |
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-ra +(q - r)b, c |
(1/p)(-rx + qy), z; +((ur/p), (uq/p), 0) |
((p - 1)/6) × 6g |
((p - 1)/6) × 6g |
((p - 1)/3) × 6g |
((p - 1)/3) × 6g |
((p - 1)/2) × 6g |
((p - 1)/2) × 6g |
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p = prime = q2 - qr + r2 = 6n + 1; |
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q, r = 1, 2, . . .; q> r; u = 1, . . ., p - 1 |
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| [4] P-3 |
2a, 2b, c |
(1/2)x, (1/2)y, z; +((1/2), 0, 0); |
1a; 3e |
1b; 3f |
2c; 6g |
2d; 6g |
2 × 6g |
2 × 6g |
4 × 6g |
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+(0, (1/2), 0); +((1/2), (1/2), 0) |
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| [p2] P-3 |
pa, pb, c |
(1/p)x, (1/p)y, z; +((u/p), (v/p), 0) |
1a; |
1b; |
2c; |
2d; |
3e; |
3f; |
p2 × 6g |
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p = prime = 6n - 1; u, v = 1, . . ., p - 1 |
((p2 - 1)/6) × 6g |
((p2 - 1)/6) × 6g |
((p2 - 1)/3) × 6g |
((p2 - 1)/3) × 6g |
((p2 - 1)/2) × 6g |
((p2 - 1)/2) × 6g |
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