P-42m No. 111 D2d1


Axes Coordinates Wyckoff positions
1a 1b 1c 1d 2e 2f 2g 2h 4i 4j 4k 4l
4m 4n 8o
I Maximal translationengleiche subgroups
[2] P-4 (81) 1a 1d 1b 1c 2g 2g 2e 2f 4h 4h 4h 4h
2 × 2g 4h 2 × 4h
[2] P222 1a 1h 1d 1e 1b; 1c 1f; 1g 2q 2t 2i; 2m 2l; 2p 2j; 2n 2k; 2o
(16) 2r; 2s 4u 2 × 4u
[2] Cmm2 a - b, (1/2)(x - y), (1/2)(x + y), z 2a 2b 2a 2b 4c 4c 2 × 2a 2 × 2b 8f 8f 8f 8f
(35) a + b, c 2 × 4c 4d; 4e 2 × 8f
II Maximal klassengleiche subgroups
   Enlarged unit cell, non-isomorphic
[2] I-4c2 a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z; 4b 4c 4a 4d 8h 8e 8f 8g 16i 16i 2 × 8e 2 × 8h
(120) a + b, 2c +(0, 0, (1/2)) 16i 16i 2 × 16i
[2] I-4c2 a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z + (1/4); 4a 4d 4b 4c 8e 8h 8f 8g 2 × 8e 2 × 8h 16i 16i
(120) a + b, 2c +(0, 0, (1/2)) 16i 16i 2 × 16i
[2] I-4m2 a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z; 2a; 2b 2c; 2d 4e 4f 8g 8h 2 × 4e 2 × 4f 2 × 8g 2 × 8h 16j 16j
(119) a + b, 2c +(0, 0, (1/2)) 16j 2 × 8i 2 × 16j
[2] I-4m2 a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z + (1/4); 4e 4f 2a; 2b 2c; 2d 8h 8g 2 × 4e 2 × 4f 16j 16j 2 × 8g 2 × 8h
(119) a + b, 2c +(0, 0, (1/2)) 16j 2 × 8i 2 × 16j
[2] P-4b2 a - b, (1/2)(x - y), (1/2)(x + y), z; 2a 2d 2b 2c 4g 4h 4e 4f 8i 2 × 4h 8i 2 × 4g
(117) a + b, c +((1/2), (1/2), 0) 8i 8i 2 × 8i
[2] P-4b2 a - b, (1/2)(x - y) + (1/2), (1/2)(x + y), z; 2c 2b 2d 2a 4g 4h 4f 4e 2 × 4g 8i 2 × 4h 8i
(117) a + b, c +((1/2), (1/2), 0) 8i 8i 2 × 8i
[2] P-4m2 a - b, (1/2)(x - y), (1/2)(x + y), z; 1a; 1b 2g 1c; 1d 2g 4h 4i 2e; 2f 2 × 2g 2 × 4h 8l 2 × 4i 8l
(115) a + b, c +((1/2), (1/2), 0) 8l 4j; 4k 2 × 8l
[2] P-4m2 a - b, (1/2)(x - y) + (1/2), (1/2)(x + y), z; 2g 1c; 1d 2g 1a; 1b 4h 4i 2 × 2g 2e; 2f 8l 2 × 4i 8l 2 × 4h
(115) a + b, c +((1/2), (1/2), 0) 8l 4j; 4k 2 × 8l
[2] P-42c a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2e 2c 2a 2f 4m 2b; 2d 4k 4l 8n 4h; 4i 4g; 4j 8n
(112) 2 × 4m 8n 2 × 8n
[2] P-42c a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2a 2f 2e 2c 2b; 2d 4m 4k 4l 4g; 4j 8n 8n 4h; 4i
(112) 2 × 4m 8n 2 × 8n
   Enlarged unit cell, isomorphic
[2] P-42m a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 1a; 1c 2h 2g 1b; 1d 2e; 2f 4m 2 × 2g 2 × 2h 4i; 4k 8o 8o 4j; 4l
2 × 4m 2 × 4n 2 × 8o
[2] P-42m a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2g 1b; 1d 1a; 1c 2h 4m 2e; 2f 2 × 2g 2 × 2h 8o 4j; 4l 4i; 4k 8o
2 × 4m 2 × 4n 2 × 8o
[3] P-42m a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 1a; 2g 1b; 2h 1c; 2g 1d; 2h 2e; 4m 2f; 4m 3 × 2g 3 × 2h 4i; 8o 4j; 8o 4k; 8o 4l; 8o
3 × 4m 3 × 4n 3 × 8o


Axes Coordinates Wyckoff positions
1a 1b 1c 1d 2e 2f
2g 2h 4i 4j 4k
4l 4m 4n 8o
[p] P-42m a, b, pc x, y, (1/p)z; +(0, 0, (u/p)) 1a; ((p - 1)/2) × 2g 1b; ((p - 1)/2) × 2h 1c; ((p - 1)/2) × 2g 1d; ((p - 1)/2) × 2h 2e; ((p - 1)/2) × 4m 2f; ((p - 1)/2) × 4m
p = prime > 2; p × 2g p × 2h 4i; ((p - 1)/2) × 8o 4j; ((p - 1)/2) × 8o 4k; ((p - 1)/2) × 8o
u = 1, . . ., p - 1 4l; ((p - 1)/2) × 8o p × 4m p × 4n p × 8o
[9] P-42m 3a, 3b, c (1/3)x, (1/3)y, z; 1a; 4i; 4n 1b; 4j; 4n 1c; 4k; 4n 1d; 4l; 4n 2e; 4i; 4l; 8o 2f; 4j; 4k; 8o
±((1/3), 0, 0); ±(0, (1/3), 0); 2g; 2 × 4n; 8o 2h; 2 × 4n; 8o 3 × 4i; 3 × 8o 3 × 4j; 3 × 8o 3 × 4k; 3 × 8o
±((1/3), (1/3), 0); ±((1/3), (2/3), 0) 3 × 4l; 3 × 8o 4m; 4 × 8o 3 × 4n; 3 × 8o 9 × 8o
[p2] P-42m pa, pb, c (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) 1a; ((p - 1)/2) × 4i; 1b; ((p - 1)/2) × 4j; 1c; ((p - 1)/2) × 4k; 1d; ((p - 1)/2) × 4l; 2e; ((p - 1)/2) × 4i; 2f; ((p - 1)/2) × 4j;
p = prime > 2; ((p - 1)/2) × 4n; ((p - 1)/2) × 4n; ((p - 1)/2) × 4n; ((p - 1)/2) × 4n; ((p - 1)/2) × 4l; ((p - 1)/2) × 4k;
u, v = 1, . . ., p - 1 (((p - 1)(p - 3))/8) × 8o (((p - 1)(p - 3))/8) × 8o (((p - 1)(p - 3))/8) × 8o (((p - 1)(p - 3))/8) × 8o (((p - 1)2)/4) × 8o (((p - 1)2)/4) × 8o
2g; (p - 1) × 4n; 2h; (p - 1) × 4n; p × 4i; p × 4j; p × 4k;
(((p - 1)2)/4) × 8o (((p - 1)2)/4) × 8o ((p(p - 1))/2) × 8o ((p(p - 1))/2) × 8o ((p(p - 1))/2) × 8o
p × 4l; 4m; ((p2 - 1)/2) × 8o p × 4n; p2 × 8o
((p(p - 1))/2) × 8o ((p(p - 1))/2) × 8o









































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