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Axes |
Coordinates |
Wyckoff positions |
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8a |
8b |
12c |
12d |
16e |
24f |
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24g |
24h |
48i |
| I Maximal translationengleiche subgroups |
| [2] I213 |
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8a |
8a |
12b |
12b |
2 × 8a |
2 × 12b |
| (199) |
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24c |
24c |
2 × 24c |
| [4] R32 |
(1/2)(-a + b + c), |
y + z - (1/4), x + z - (1/4), x + y - (1/4) |
1a; 3d |
1b; 3e |
3d; 3e |
3d; 3e |
2c; 6f |
2 × 6f |
| (155) |
(1/2)(a - b + c), |
(rhombohedral |
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2 × 3e; 6f |
2 × 3d; 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), (1/3)(-x - y + 2z), |
3a; 9d |
3b; 9e |
9d; 9e |
9d; 9e |
6c; 18f |
2 × 18f |
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(1/2)(a + b + c) |
(2/3)(x + y + z) - (1/4) |
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2 × 9e; 18f |
2 × 9d; 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) + (1/2), |
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(1/2)(-a + b - c) |
(2/3)(-x + y - z) - (1/4) |
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(hexagonal axes) |
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| conjugate: |
-a - b, b - c, |
(1/3)(-2x - y - z) + (1/2), (1/3)(-x + y - 2z), |
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(1/2)(a - b - c) |
(2/3)(x - y - z) - (1/4) |
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(hexagonal axes) |
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| conjugate: |
a - b, b + c, |
(1/3)(2x - y + z) + (1/2), (1/3)(x + y + 2z) + (1/2), |
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(1/2)(-a - b + c) |
(2/3)(-x - y + z) - (1/4) |
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(hexagonal axes) |
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| [3] I4122 (98) |
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x, y - (1/4), z - (1/8) |
8e |
8d |
4a; 8f |
4b; 8f |
16g |
8c; 2 × 8f |
| conjugate: |
b, c, a |
y, z - (1/4), x - (1/8) |
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8d; 16g |
8e; 16g |
3 × 16g |
| conjugate: |
c, a, b |
z, x - (1/4), y - (1/8) |
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| II Maximal klassengleiche subgroups |
| Loss of centring translations |
| [2] P4132 |
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8c |
4a; 4b |
12d |
12d |
2 × 8c |
24e |
| (213) |
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2 × 12d |
24e |
2 × 24e |
| [2] P4332 |
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4a; 4b |
8c |
12d |
12d |
2 × 8c |
24e |
| (212) |
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24e |
2 × 12d |
2 × 24e |
| Enlarged unit cell, isomorphic |
| [27] I4132 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
8b; 16e; |
8a; 16e; |
12c; 24f; |
12d; 24f; |
3 × 16e; |
3 × 24f; |
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±(0, (1/3), 0); ±(0, 0, (1/3)); |
2 × 24g; |
2 × 24h; |
24g; 24h; |
24g; 24h; |
8 × 48i |
12 × 48i |
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±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
3 × 48i |
3 × 48i |
5 × 48i |
5 × 48i |
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±((1/3), (1/3), 0); ±((1/3), (1/3), (1/3)); |
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±(0, (1/3), (2/3)); ±((1/3), 0, (2/3)); |
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3 × 24h; |
3 × 24g; |
27 × 48i |
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±((1/3), (2/3), 0); ±((1/3), (1/3), (2/3)); |
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12 × 48i |
12 × 48i |
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±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) |
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| [p3] I4132 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
8a(b*); |
8b(a*); |
12c(d† ); |
12d(c† ); |
p × 16e; |
p × 24f; |
|
p = prime > 2; u, v, w = 1, . . ., p - 1 |
((p - 1)/2) × 16e; |
((p - 1)/2) × 16e; |
((p - 1)/2) × 24f; |
((p - 1)/2) × 24f; |
((p(p2 - 1))/3) × 48i |
((p(p2 - 1))/2) × 48i |
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(p - 1) × |
(p - 1) × |
((p - 1)/2) × 24g; |
((p - 1)/2) × 24g; |
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24h(g*); |
24g(h*); |
((p - 1)/2) × 24h; |
((p - 1)/2) × 24h; |
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((p3 - 4p + 3)/6) × |
((p3 - 4p + 3)/6) × |
(((p - 1)2(p + 2))/4) × |
(((p - 1)2(p + 2))/4) × |
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48i |
48i |
48i |
48i |
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p × 24g(h*); |
p × 24h(g*); |
p3 × 48i |
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((p(p2 - 1))/2) × 48i |
((p(p2 - 1))/2) × 48i |
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| * p = 4n - 1 |
† p = 8n + 5 or p = 8n + 7 |
