|
Axes |
Coordinates |
Wyckoff positions |
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1a |
1b |
3c |
3d |
4e |
6f |
6g |
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6h |
6i |
12j |
| I Maximal translationengleiche subgroups |
| [4] R3 |
(rhombohedral axes) |
1a |
1a |
3b |
3b |
1a; 3b |
2 × 3b |
2 × 3b |
| (146) |
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2 × 3b |
2 × 3b |
4 × 3b |
|
a - b, b - c, |
(1/3)(2x - y - z), (1/3)(x + y - 2z), |
3a |
3a |
9b |
9b |
3a; 9b |
2 × 9b |
2 × 9b |
|
a + b + c |
(1/3)(x + y + z) |
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2 × 9b |
2 × 9b |
4 × 9b |
|
(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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| 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] P222 |
|
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1a |
1h |
1e; 1f; 1g |
1b; 1c; 1d |
4u |
2i; 2m; 2q |
2j; 2o; 2s |
| (16) |
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2k; 2n; 2r |
2l; 2p; 2t |
3 × 4u |
| II Maximal klassengleiche subgroups |
| Enlarged unit cell, non-isomorphic |
| [2] F23 |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0) |
4a; 4b |
4c; 4d |
24g |
24f |
2 × 16e |
2 × 24f |
48h |
| (196) |
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48h |
2 × 24g |
2 × 48h |
| [4] I213 |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0); |
8a |
8a |
2 × 12b |
2 × 12b |
8a; 24c |
2 × 24c |
4 × 12b |
| (199) |
|
+(0, (1/2), 0); +(0, 0, (1/2)) |
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2 × 24c |
2 × 24c |
4 × 24c |
| [4] I213 |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); +((1/2), 0, 0); |
8a |
8a |
2 × 12b |
2 × 12b |
8a; 24c |
2 × 24c |
2 × 24c |
| (199) |
|
+(0, (1/2), 0); +(0, 0, (1/2)) |
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4 × 12b |
2 × 24c |
4 × 24c |
| [4] I23 |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0); |
2a; 6b |
8c |
24f |
12d; 12e |
8c; 24f |
2 × 12d; 2 × 12e |
2 × 24f |
| (197) |
|
+(0, (1/2), 0); +(0, 0, (1/2)) |
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2 × 24f |
2 × 24f |
4 × 24f |
| [4] I23 |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); +((1/2), 0, 0); |
8c |
2a; 6b |
12d; 12e |
24f |
8c; 24f |
2 × 24f |
2 × 24f |
| (197) |
|
+(0, (1/2), 0); +(0, 0, (1/2)) |
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2 × 24f |
2 × 12d; 2 × 12e |
4 × 24f |
|
Axes |
Coordinates |
Wyckoff positions |
|
|
|
1a |
1b |
3c |
3d |
4e |
6f |
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6g |
6h |
6i |
12j |
| Enlarged unit cell, isomorphic |
| [27] P23 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
1a; 2 × 4e; 6f; |
1b; 2 × 4e; 6i; |
3c; 6g; 6h; |
3d; 6f; 6g; |
3 × 4e; |
3 × 6f; |
|
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±(0, (1/3), 0); ±(0, 0, (1/3)); |
12j |
12j |
6i; 5 × 12j |
6h; 5 × 12j |
8 × 12j |
12 × 12j |
|
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±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
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±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
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3 × 6g; |
3 × 6h; |
3 × 12i; |
27 × 12j |
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±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
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12 × 12j |
12 × 12j |
12 × 12j |
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±((1/3), (1/3), (1/3)); ±((1/3), (1/3), (2/3)); |
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±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) |
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| [p3] P23 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; |
1a; (p - 1) × 4e; |
1b; (p - 1) × 4e; |
3c; ((p - 1)/2) × 6g; |
3d; ((p - 1)/2) × 6f; |
p × 4e; |
p × 6f; |
|
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+((u/p), (v/p), (w/p)) |
((p - 1)/2) × 6f; |
((p - 1)/2) × 6i; |
((p - 1)/2) × 6h; |
((p - 1)/2) × 6g; |
((p(p2 - 1))/3) × 12j |
((p(p2 - 1))/2) × 12j |
|
p = prime > 2; |
(((p - 1)(p - 2)(p + 3))/12) |
(((p - 1)(p - 2)(p + 3))/12) |
((p - 1)/2) × 6i; |
((p - 1)/2) × 6h; |
|
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u, v, w = 1, . . ., p - 1 |
× 12j |
× 12j |
(((p - 1)2(p + 2))/4) × |
(((p - 1)2(p + 2))/4) × |
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12j |
12j |
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p × 6g; |
p × 6h; |
p × 6i; |
p3 × 12j |
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((p(p2 - 1))/2) × 12j |
((p(p2 - 1))/2) × 12j |
((p(p2 - 1))/2) × 12j |
|
