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Axes |
Coordinates |
Wyckoff positions |
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2a |
6b |
8c |
12d |
12e |
16f |
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24g |
24h |
24i |
48j |
| I Maximal translationengleiche subgroups |
| [2] I23 |
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2a |
6b |
8c |
12e |
12d |
2 × 8c |
| (197) |
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2 × 12e |
24f |
24f |
2 × 24f |
| [4] R32 |
(1/2)(-a + b + c), |
y + z, x + z, x + y |
1a |
3d |
1b; 3e |
2 × 3e |
6f |
2c; 6f |
| (155) |
(1/2)(a - b + c), |
(rhombohedral |
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2 × 6f |
2 × 3d; 6f |
2 × 3e; 6f |
4 × 6f |
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(1/2)(a + b - c) |
axes) |
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-a + b, -b + c, |
(1/3)(-2x + y + z), |
3a |
9d |
3b; 9e |
2 × 9e |
18f |
6c; 18f |
|
(1/2)(a + b + c) |
(1/3)(-x - y + 2z), (2/3)(x + y + z) |
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2 × 18f |
2 × 9d; 18f |
2 × 9e; 18f |
4 × 18f |
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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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(1/2)(-a + b - c) |
(2/3)(-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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(1/2)(a - b - c) |
(2/3)(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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(1/2)(-a - b + c) |
(2/3)(-x - y + z) |
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(hexagonal axes) |
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| [3] I422 (97) |
|
2a |
2b; 4c |
8j |
4d; 8i |
4e; 8h |
16k |
| conjugate: |
b, c, a |
y, z, x |
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8f; 2 × 8i |
8g; 16k |
8j; 16k |
3 × 16k |
| conjugate: |
c, a, b |
z, x, y |
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| II Maximal klassengleiche subgroups |
| Loss of centring translations |
| [2] P4232 |
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2a |
6d |
4b; 4c |
6e; 6f |
12h |
2 × 8g |
| (208) |
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12i; 12j |
24m |
12k; 12l |
2 × 24m |
| [2] P432 |
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1a; 1b |
3c; 3d |
8g |
12h |
6e; 6f |
2 × 8g |
| (207) |
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2 × 12h |
12i; 12j |
24k |
2 × 24k |
| Enlarged unit cell, isomorphic |
| [27] I432 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
2a; 12e; |
6b; 12e; 24g; |
8c; 16f; |
12d; 24g; |
3 × 12e; |
3 × 16f; |
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±(0, (1/3), 0); ±(0, 0, (1/3)); |
16f; 24h |
24h; 2 × 48j |
2 × 24i; |
2 × 24i; |
6 × 48j |
8 × 48j |
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±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
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3 × 48j |
5 × 48j |
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±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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3 × 24g; |
3 × 24h; |
3 × 24i; |
27 × 48j |
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±((1/3), (1/3), (1/3)); ±((2/3), (1/3), (1/3)); |
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12 × 48j |
12 × 48j |
12 × 48j |
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±((1/3), (2/3), (1/3)); ±((1/3), (1/3), (2/3)) |
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| [p3] I432 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
2a; |
6b; |
8c; |
12d; |
p × 12e; |
p × 16f; |
|
p = prime > 2; u, v, w = 1, . . ., p - 1 |
((p - 1)/2) × 12e; |
((p - 1)/2) × 12e; |
((p - 1)/2) × 16f; |
((p - 1)/2) × 24g; |
((p(p2 - 1))/4) × 48j |
((p(p2 - 1))/3) × 48j |
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((p - 1)/2) × 16f; |
((p - 1)/2) × 24g; |
(p - 1) × 24i; |
(p - 1) × 24i; |
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((p - 1)/2) × 24h; |
((p - 1)/2) × 24h; |
((p3 - 4p + 3)/6) × |
(((p - 1)2(p + 2))/4) × |
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((p3 - 13p + 12)/24) × |
(((p3 - 5p + 4))/8) × |
48j |
48j |
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48j |
48j |
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p × 24g; |
p × 24h; |
p × 24i; |
p3 × 48j |
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((p(p2 - 1))/2) × 48j |
((p(p2 - 1))/2) × 48j |
((p(p2 - 1))/2) × 48j |
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