|
Axes |
Coordinates |
Wyckoff positions |
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|
1a |
1b |
3c |
3d |
4e |
6f |
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8g |
12h |
12i |
12j |
24k |
| I Maximal translationengleiche subgroups |
| [2] P23 |
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1a |
1b |
3c |
3d |
6f |
6i |
| (195) |
|
|
|
2 × 4e |
6g; 6h |
12j |
12j |
2 × 12j |
| [4] R32 |
(rhombohedral axes) |
1a |
1b |
3d |
3e |
6f |
6f |
| (155) |
|
|
|
2c; 6f |
2 × 6f |
2 × 3d; 6f |
2 × 3e; 6f |
4 × 6f |
| (hexagonal |
a - b, b - c, |
(1/3)(2x - y - z), (1/3)(x + y - 2z), |
3a |
3b |
9d |
9e |
18f |
18f |
| axes) |
a + b + c |
(1/3)(x + y + z) |
|
6c; 18f |
2 × 18f |
2 × 9d; 18f |
2 × 9e; 18f |
4 × 18f |
| conjugate: |
-a - b, b + c, |
(1/3)(-2x - y + z), (1/3)(-x + y + 2z), |
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-a + b - c |
(1/3)(-x + y - z) |
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(hexagonal axes) |
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| conjugate: |
a + b, -b + c, |
(1/3)(2x + y + z), (1/3)(x - y + 2z), |
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a - b - c |
(1/3)(x - y - z) |
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(hexagonal axes) |
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| conjugate: |
-a + b, -b - c, |
(1/3)(-2x + y - z), (1/3)(-x - y - 2z), |
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-a - b + c |
(1/3)(-x - y + z) |
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(hexagonal axes) |
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| [3] P422(89) |
|
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1a |
1d |
1c; 2f |
1b; 2e |
2g; 4l |
2h; 4m |
| conjugate: |
b, c, a |
y, z, x |
|
8p |
4i; 4n; 4o |
4j; 8p |
4k; 8p |
3 × 8p |
| conjugate: |
c, a, b |
z, x, y |
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| II Maximal klassengleiche subgroups |
| Enlarged unit cell, non-isomorphic |
| [2] F432 |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0) |
4a; 4b |
8c |
24d |
24e |
2 × 24e |
48i |
| (209) |
|
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|
2 × 32f |
96j |
48g; 48h |
96j |
2 × 96j |
| [2] F432 |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); |
8c |
4a; 4b |
24e |
24d |
48i |
2 × 24e |
| (209) |
|
+((1/2), 0, 0) |
|
2 × 32f |
96j |
96j |
48g; 48h |
2 × 96j |
| [4] I432 |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0); |
2a; 6b |
8c |
24h |
12d; 12e |
2 × 12e; 24g |
48j |
| (211) |
|
+(0, (1/2), 0); +(0, 0, (1/2)) |
|
16f; 48j |
2 × 48j |
2 × 24h; 48j |
2 × 24i; 48j |
4 × 48j |
| [4] I432 |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); |
8c |
2a; 6b |
12d; 12e |
24h |
48j |
2 × 12e; 24g |
| (211) |
|
+((1/2), 0, 0); +(0, (1/2), 0); +(0, 0, (1/2)) |
|
16f; 48j |
2 × 48j |
2 × 24i; 48j |
2 × 24h; 48j |
4 × 48j |
| Enlarged unit cell, isomorphic |
| [27] P432 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
1a; 6e; 8g; |
1b; 6f; 8g; |
3c; 6f; 12h; |
3d; 6e; 12h; |
3 × 6e; |
3 × 6f; |
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±(0, (1/3), 0); ±(0, 0, (1/3)); ±(0, (1/3), (1/3)); |
12i |
12j |
12i; 2 × 24k |
12j; 2 × 24k |
6 × 24k |
6 × 24k |
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±((1/3), 0, (1/3)); ±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); ±((1/3), (1/3), (1/3)); |
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3 × 8g; |
3 × 12h; |
3 × 12i; |
3 × 12j; |
27 × 24k |
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±((1/3), (1/3), (2/3)); ±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) |
|
8 × 24k |
12 × 24k |
12 × 24k |
12 × 24k |
|
| [p3] P432 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
1a; ((p - 1)/2) × 6e; |
1b; ((p - 1)/2) × 6f; |
3c; ((p - 1)/2) × 6f; |
3d; ((p - 1)/2) × 6e; |
p × 6e; |
p × 6f; |
|
p = prime > 2; u, v, w = 1, . . ., p - 1 |
((p - 1)/2) × 8g; |
((p - 1)/2) × 8g; |
((p - 1)/2) × 12h; |
((p - 1)/2) × 12h; |
((p(p2 - 1))/4) × 24k |
((p(p2 - 1))/4) × |
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((p - 1)/2) × 12i; |
((p - 1)/2) × 12j; |
((p - 1)/2) × 12i; |
((p - 1)/2) × 12j; |
|
24k |
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((p3 - 13p + 12)/24) × |
((p3 - 13p + 12)/24) × |
((p3 - 5p + 4)/8) × |
((p3 - 5p + 4)/8) × |
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24k |
24k |
24k |
24k |
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p × 8g; |
p × 12h; |
p × 12i; |
p × 12j; |
p3 × 24k |
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((p(p2 - 1))/3) × 24k |
((p(p2 - 1))/2) × 24k |
((p(p2 - 1))/2) × 24k |
((p(p2 - 1))/2) × 24k |
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