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Axes |
Coordinates |
Wyckoff positions |
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origin 1 |
origin 2 |
2a |
2b |
4c |
4d |
4e |
4f |
8g |
| I Maximal translationengleiche subgroups |
| [2] P42 (77) |
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x + (1/2), y, z |
x + (1/4), y - (1/4), z |
2c |
2c |
4d |
4d |
2a; 2b |
2 × 2c |
2 × 4d |
| [2] P-4 (81) |
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x + (1/4), y + (1/4), z + (1/4) |
1a; 1d |
1b; 1c |
4h |
4h |
2 × 2g |
2e; 2f |
2 × 4h |
| [2] P112/n (13) |
x - (1/4), y - (1/4), z - (1/4) |
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2e |
2e |
2a; 2b |
2c; 2d |
2 × 2f |
2 × 2e |
2 × 4g |
| II Maximal klassengleiche subgroups |
| Enlarged unit cell, non-isomorphic |
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| [2] I41/a |
a - b, |
(1/2)(x - y), (1/2)(x + y), (1/2)z; +(0, 0, (1/2)) |
4a; 4b |
8e |
8c; 8d |
16f |
16f |
2 × 8e |
2 × 16f |
| (88) |
a + b, 2c |
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| [2] I41/a |
a - b, |
(1/2)(x - y), (1/2)(x + y), (1/2)z + (1/4); +(0, 0, (1/2)) |
8e |
4a; 4b |
16f |
8c; 8d |
16f |
2 × 8e |
2 × 16f |
| (88) |
a + b, 2c |
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| [2] I41/a |
a - b, |
(1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z; +(0, 0, (1/2)) |
8e |
4a; 4b |
8c; 8d |
16f |
16f |
2 × 8e |
2 × 16f |
| (88) |
a + b, 2c |
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| [2] I41/a |
a - b, |
(1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z + (1/4); |
4a; 4b |
8e |
16f |
8c; 8d |
16f |
2 × 8e |
2 × 16f |
| (88) |
a + b, 2c |
+(0, 0, (1/2)) |
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| Enlarged unit cell, isomorphic |
| [3] P42/n |
a, b, 3c |
x, y, (1/3)z; |
x, y, (1/3)z; |
2a(b*); 4f |
2b(a*); 4f |
4d(c*); 8g |
4c(d*); 8g |
3 × 4e |
3 × 4f |
3 × 8g |
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±(0, 0, (1/3)) |
±(0, 0, (1/3)) |
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| [p] P42/n |
a, b, pc |
x, y, (1/p)z; |
x, y, (1/p)z; |
2a(b† ); |
2b(a† ); |
4c(d‡ ); |
4d(c‡ ); |
p × 4e |
p × 4f |
p × 8g |
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+(0, 0, (u/p)) |
+(0, 0, (u/p)) |
((p - 1)/2) × 4f |
((p - 1)/2) × 4f |
((p - 1)/2) × 8g |
((p - 1)/2) × 8g |
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p = prime > 2; u = 1, . . ., p - 1 |
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| [5] P42/n |
a + 2b, |
(1/5)(x + 2y), |
(1/5)(x + 2y), |
2a(b*); 8g |
2b(a*); 8g |
4c; 2 × 8g |
4d; 2 × 8g |
4e; 2 × 8g |
4f; 2 × 8g |
5 × 8g |
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-2a + b, c |
(1/5)(-2x + y), z; |
(1/5)(-2x + y), z; |
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±((1/5), (3/5), 0); ±((2/5), (1/5), 0) |
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| [5] P42/n |
a - 2b, |
(1/5)(x - 2y), |
(1/5)(x - 2y), |
2a(b*); 8g |
2b(a*); 8g |
4c; 2 × 8g |
4d; 2 × 8g |
4e; 2 × 8g |
4f; 2 × 8g |
5 × 8g |
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2a + b, c |
(1/5)(2x + y), z; |
(1/5)(2x + y), z; |
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±((1/5), (2/5), 0); ±((3/5), (1/5), 0) |
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| [p] P42/n |
qa - rb, |
(1/p)(qx - ry), |
(1/p)(qx - ry), |
2a(b**); |
2b(a**); |
4c; |
4d; |
4e; |
4f; |
p × 8g |
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ra + qb, c |
(1/p)(rx + qy), z; |
(1/p)(rx + qy), z; |
((p - 1)/4) × 8g |
((p - 1)/4) × 8g |
((p - 1)/2) × 8g |
((p - 1)/2) × 8g |
((p - 1)/2) × 8g |
((p - 1)/2) × 8g |
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+((uq/p), (ur/p), 0) |
+((uq/p), (ur/p), 0) |
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p = q2 + r2 = prime = 4n + 1; |
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q = 2n + 1≥ 1; r = ± 2n' ≠ 0; u = 1, . . ., p - 1 |
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| [9] P42/n |
3a, 3b, c |
(1/3)x, (1/3)y, z; |
(1/3)x, (1/3)y, z; |
2a(b**); |
2b(a**); |
4c; 4 × 8g |
4d; 4 × 8g |
4e; 4 × 8g |
4f; 4 × 8g |
9 × 8g |
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±((1/3), 0, 0); ±(0, (1/3), 0); |
2 × 8g |
2 × 8g |
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±((1/3), (1/3), 0); ±((1/3), (2/3), 0) |
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| [p2] P42/n |
pa, pb, c |
(1/p)x, (1/p)y, z; |
(1/p)x, (1/p)y, z; |
2a(b† ); |
2b(a† ); |
4c; |
4d; |
4e; |
4f; |
p2 × 8g |
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+((u/p), (v/p), 0) |
+((u/p), (v/p), 0) |
((p2 - 1)/4) × 8g |
((p2 - 1)/4) × 8g |
((p2 - 1)/2) × 8g |
((p2 - 1)/2) × 8g |
((p2 - 1)/2) × 8g |
((p2 - 1)/2) × 8g |
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p = prime = 4n - 1; u, v = 1, . . ., p - 1 |
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| * origin 2 |
| † origin 2 and p = 4n - 1 |
| ‡ origin 1 and p = 4n - 1 |
| ** origin 2 and q + r = 4n - 1 |
