P422 | No. 89 | D41 |
Axes | Coordinates | Wyckoff positions | |||||||||||
1a | 1b | 1c | 1d | 2e | 2f | 2g | 2h | 4i | 4j | 4k | |||
4l | 4m | 4n | 4o | 8p | |||||||||
I Maximal translationengleiche subgroups | |||||||||||||
[2] P4 (75) | 1a | 1a | 1b | 1b | 2c | 2c | 2 × 1a | 2 × 1b | 2 × 2c | 4d | 4d | ||
4d | 4d | 4d | 4d | 2 × 4d | |||||||||
[2] P222 (16) | 1a | 1d | 1e | 1h | 1b; 1c | 1f; 1g | 2q | 2t | 2r; 2s | 4u | 4u | ||
2i; 2m | 2l; 2p | 2j; 2n | 2k; 2o | 2 × 4u | |||||||||
[2] C222 | a - b, a + b, c | (1/2)(x - y), (1/2)(x + y), z | 2a | 2d | 2b | 2c | 4k | 4k | 4i | 4j | 2 × 4k | 4e; 4g | 4f; 4h |
(21) | 8l | 8l | 8l | 8l | 2 × 8l | ||||||||
II Maximal klassengleiche subgroups | |||||||||||||
Enlarged unit cell, non-isomorphic | |||||||||||||
[2] I422 | a - b, a + b, 2c | (1/2)(x - y), (1/2)(x + y), (1/2)z; | 2a; 2b | 4e | 4c | 4d | 8g | 8j | 2 × 4e | 8f | 16k | 8h; 8i | 16k |
(97) | +(0, 0, (1/2)) | 2 × 8g | 2 × 8j | 16k | 16k | 2 × 16k | |||||||
[2] I422 | a - b, a + b, 2c | (1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z; | 4c | 4d | 2a; 2b | 4e | 8g | 8j | 8f | 2 × 4e | 16k | 8h; 8i | 16k |
(97) | +(0, 0, (1/2)) | 16k | 16k | 2 × 8j | 2 × 8g | 2 × 16k | |||||||
[2] I422 | a - b, a + b, 2c | (1/2)(x - y), (1/2)(x + y), | 4e | 2a; 2b | 4d | 4c | 8j | 8g | 2 × 4e | 8f | 16k | 16k | 8h; 8i |
(97) | (1/2)z + (1/4); +(0, 0, (1/2)) | 16k | 16k | 2 × 8g | 2 × 8j | 2 × 16k | |||||||
[2] I422 | a - b, a + b, 2c | (1/2)(x - y) + (1/2), (1/2)(x + y), | 4d | 4c | 4e | 2a; 2b | 8j | 8g | 8f | 2 × 4e | 16k | 16k | 8h; 8i |
(97) | (1/2)z + (1/4); +(0, 0, (1/2)) | 2 × 8j | 2 × 8g | 16k | 16k | 2 × 16k | |||||||
[2] P4222 | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) | 2a | 2e | 2b | 2f | 2c; 2d | 4i | 4g | 4h | 2 × 4i | 8p | 4n; 4o |
(93) | 4j; 4l | 8p | 8p | 4k; 4m | 2 × 8p | ||||||||
[2] P4222 | a, b, 2c | x, y, (1/2)z + (1/4); +(0, 0, (1/2)) | 2e | 2a | 2f | 2b | 4i | 2c; 2d | 4g | 4h | 2 × 4i | 4n; 4o | 8p |
(93) | 8p | 4k; 4m | 4j; 4l | 8p | 2 × 8p | ||||||||
[2] P4212 | a - b, a + b, c | (1/2)(x - y), (1/2)(x + y), z; | 2a | 2b | 2c | 2c | 4e | 4f | 4d | 2 × 2c | 8g | 8g | 8g |
(90) | +((1/2), (1/2), 0) | 2 × 4e | 8g | 2 × 4f | 8g | 2 × 8g | |||||||
[2] P4212 | a - b, a + b, c | (1/2)(x - y) + (1/2), (1/2)(x + y), z; | 2c | 2c | 2a | 2b | 4e | 4f | 2 × 2c | 4d | 8g | 8g | 8g |
(90) | +((1/2), (1/2), 0) | 8g | 2 × 4f | 8g | 2 × 4e | 2 × 8g | |||||||
Enlarged unit cell, isomorphic | |||||||||||||
[2] P422 | a, b, 2c | x, y, (1/2)z; +(0, 0, (1/2)) | 1a; 1b | 2g | 1c; 1d | 2h | 2e; 2f | 4i | 2 × 2g | 2 × 2h | 2 × 4i | 4j; 4k | 8p |
4l; 4n | 8p | 8p | 4m; 4o | 2 × 8p | |||||||||
[2] P422 | a, b, 2c | x, y, (1/2)z + (1/4); +(0, 0, (1/2)) | 2g | 1a; 1b | 2h | 1c; 1d | 4i | 2e; 2f | 2 × 2g | 2 × 2h | 2 × 4i | 8p | 4j; 4k |
8p | 4m; 4o | 4l; 4n | 8p | 2 × 8p | |||||||||
[3] P422 | a, b, 3c | x, y, (1/3)z; ±(0, 0, (1/3)) | 1a; 2g | 1b; 2g | 1c; 2h | 1d; 2h | 2e; 4i | 2f; 4i | 3 × 2g | 3 × 2h | 3 × 4i | 4j; 8p | 4k; 8p |
4l; 8p | 4m; 8p | 4n; 8p | 4o; 8p | 3 × 8p |
Axes | Coordinates | Wyckoff positions | |||||||
1a | 1b | 1c | 1d | 2e | 2f | 2g | |||
2h | 4i | 4j | 4k | 4l | 4m | ||||
4n | 4o | 8p | |||||||
[p] P422 | a, b, pc | x, y, (1/p)z; | 1a; ((p - 1)/2) × 2g | 1b; ((p - 1)/2) × 2g | 1c; ((p - 1)/2) × 2h | 1d; ((p - 1)/2) × 2h | 2e; ((p - 1)/2) × 4i | 2f; ((p - 1)/2) × 4i | p × 2g |
+(0, 0, (u/p)) | p × 2h | p × 4i | 4j; ((p - 1)/2) × 8p | 4k; ((p - 1)/2) × 8p | 4l; ((p - 1)/2) × 8p | 4m; ((p - 1)/2) × 8p | |||
p = prime > 2; | 4n; ((p - 1)/2) × 8p | 4o; ((p - 1)/2) × 8p | p × 8p | ||||||
u = 1, . . ., p - 1 | |||||||||
[2] P422 | a - b, | (1/2)(x - y), | 1a; 1c | 1b; 1d | 2e | 2f | 4j | 4k | 2g; 2h |
a + b, c | (1/2)(x + y), z; | 4i | 8p | 4l; 4o | 4m; 4n | 2 × 4j | 8p | ||
+((1/2), (1/2), 0) | 2 × 4k | 8p | 2 × 8p | ||||||
[2] P422 | a - b, | (1/2)(x - y) + (1/2), | 2e | 2f | 1a; 1c | 1b; 1d | 4j | 4k | 4i |
a + b, c | (1/2)(x + y), z; | 2g; 2h | 8p | 4l; 4o | 4m; 4n | 8p | 2 × 4k | ||
+((1/2), (1/2), 0) | 8p | 2 × 4j | 2 × 8p | ||||||
[9] P422 | 3a, 3b, c | (1/3)x, (1/3)y, z; | 1a; 4j; 4l | 1b; 4k; 4n | 1c; 4j; 4o | 1d; 4k; 4m | 2e; 4l; 4o; 8p | 2f; 4m; 4n; 8p | 2g; 2 × 8p |
±((1/3), 0, 0); | 2h; 2 × 8p | 4i; 4 × 8p | 3 × 4j; 3 × 8p | 3 × 4k; 3 × 8p | 3 × 4l; 3 × 8p | 3 × 4m; 3 × 8p | |||
±(0, (1/3), 0); | 3 × 4n; 3 × 8p | 3 × 4o; 3 × 8p | 9 × 8p | ||||||
±((1/3), (1/3), 0); | |||||||||
±((1/3), (2/3), 0) | |||||||||
[p2] P422 | pa, pb, c | (1/p)x, (1/p)y, z; | 1a; ((p - 1)/2) × 4j; | 1b; ((p - 1)/2) × 4k; | 1c; ((p - 1)/2) × 4j; | 1d; ((p - 1)/2) × 4k; | 2e; ((p - 1)/2) × 4l; | 2f; ((p - 1)/2) × 4m; | 2g; ((p2 - 1)/4) × 8p |
+((u/p), (v/p), 0) | ((p - 1)/2) × 4l; | ((p - 1)/2) × 4n; | ((p - 1)/2) × 4o; | ((p - 1)/2) × 4m; | ((p - 1)/2) × 4o; | ((p - 1)/2) × 4n; | |||
p = prime > 2; | (((p - 1)(p - 3))/8) × 8p | (((p - 1)(p - 3))/8) × 8p | (((p - 1)(p - 3))/8) × 8p | (((p - 1)(p - 3))/8) × 8p | (((p - 1)2)/4) × 8p | (((p - 1)2)/4) × 8p | |||
u, v = 1, . . ., p - 1 | 2h; ((p2 - 1)/4) × 8p | 4i; ((p2 - 1)/2) × 8p | p × 4j; | p × 4k; | p × 4l; | p × 4m; | |||
((p(p - 1))/2) × 8p | ((p(p - 1))/2) × 8p | ((p(p - 1))/2) × 8p | ((p(p - 1))/2) × 8p | ||||||
p × 4n; | p × 4o; | p2 × 8p | |||||||
((p(p - 1))/2) × 8p | ((p(p - 1))/2) × 8p |