P4/mmm No. 123 P4/m2/m2/m D4h1


Axes Coordinates Wyckoff positions
1a 1b 1c 1d 2e 2f 2g 2h 4i 4j 4k
4l 4m 4n 4o 8p 8q 8r
8s 8t 16u
I Maximal translationengleiche subgroups
[2] P-4m2 (115) 1a 1d 1b 1c 2g 2g 2e 2f 2 × 2g 4h 4i
4j 4j 4k 4k 8l 8l 8l
2 × 4j 2 × 4k 2 × 8l
[2] P-42m (111) 1a 1c 1d 1b 2f 2e 2g 2h 4m 4n 4n
4i 4k 4l 4j 8o 8o 2 × 4n
8o 8o 2 × 8o
[2] P4mm (99) 1a 1a 1b 1b 2c 2c 2 × 1a 2 × 1b 2 × 2c 4d 4d
4e 4e 4f 4f 8g 8g 2 × 4d
2 × 4e 2 × 4f 2 × 8g
[2] P422 (89) 1a 1b 1c 1d 2f 2e 2g 2h 4i 4j 4k
4l 4n 4o 4m 8p 8p 8p
8p 8p 2 × 8p
[2] P4/m (83) 1a 1b 1c 1d 2f 2e 2g 2h 4i 4j 4k
4j 4k 4j 4k 2 × 4j 2 × 4k 8l
8l 8l 2 × 8l
[2] Pmmm (47) 1a 1c 1f 1h 1d; 1g 1b; 1e 2q 2t 2r; 2s 4y 4z
2i; 2m 2j; 2n 2k; 2o 2l; 2p 2 × 4y 2 × 4z 8[alpha]
4u; 4w 4v; 4x 2 × 8[alpha]
[2] Cmmm a - b, (1/2)(x - y), (1/2)(x + y), z 2a 2d 2b 2c 4f 4e 4k 4l 8m 4g; 4i 4h; 4j
(65) a + b, c 8p 8q 8p 8q 2 × 8p 2 × 8q 8n; 8o
16r 16r 2 × 16r
II Maximal klassengleiche subgroups
   Enlarged unit cell, non-isomorphic
[2] I4/mcm a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z; 4c 4a 4d 4b 8e 8h 8f 8g 16l 16k 16j
(140) a + b, 2c +(0, 0, (1/2)) 16k 16i 2 × 8h 16l 2 × 16k 32m 32m
32m 2 × 16l 2 × 32m
[2] I4/mcm a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z + (1/4); 4a 4c 4b 4d 8h 8e 8f 8g 16l 16j 16k
(140) a + b, 2c +(0, 0, (1/2)) 16i 16k 16l 2 × 8h 32m 2 × 16k 32m
32m 2 × 16l 2 × 32m
[2] I4/mcm a - b, (1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z; 4d 4b 4c 4a 8e 8h 8g 8f 16l 16k 16j
(140) a + b, 2c +(0, 0, (1/2)) 2 × 8h 16l 16k 16i 2 × 16k 32m 32m
2 × 16l 32m 2 × 32m
[2] I4/mcm a - b, (1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z + (1/4); 4b 4d 4a 4c 8h 8e 8g 8f 16l 16j 16k
(140) a + b, 2c +(0, 0, (1/2)) 16l 2 × 8h 16i 16k 32m 2 × 16k 32m
2 × 16l 32m 2 × 32m
[2] I4/mmm a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z; 2a; 2b 4e 4c 4d 8f 8h 2 × 4e 8g 16m 8i; 8j 16n
(139) a + b, 2c +(0, 0, (1/2)) 2 × 8h 16m 16l 16k 2 × 16l 32o 2 × 16n
2 × 16m 32o 2 × 32o
[2] I4/mmm a - b, (1/2)(x - y), (1/2)(x + y), (1/2)z + (1/4); 4e 2a; 2b 4d 4c 8h 8f 2 × 4e 8g 16m 16n 8i; 8j
(139) a + b, 2c +(0, 0, (1/2)) 16m 2 × 8h 16k 16l 32o 2 × 16l 2 × 16n
2 × 16m 32o 2 × 32o
[2] I4/mmm a - b, (1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z; 4c 4d 2a; 2b 4e 8f 8h 8g 2 × 4e 16m 8i; 8j 16n
(139) a + b, 2c +(0, 0, (1/2)) 16l 16k 2 × 8h 16m 2 × 16l 32o 2 × 16n
32o 2 × 16m 2 × 32o
[2] I4/mmm a - b, (1/2)(x - y) + (1/2), (1/2)(x + y), (1/2)z + (1/4); 4d 4c 4e 2a; 2b 8h 8f 8g 2 × 4e 16m 16n 8i; 8j
(139) a + b, 2c +(0, 0, (1/2)) 16k 16l 16m 2 × 8h 32o 2 × 16l 2 × 16n
32o 2 × 16m 2 × 32o
[2] P42/mcm a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2a 2b 2c 2d 4e 4f 4g 4h 8k 4i; 4j 8o
(132) 8n 8l 8n 8m 2 × 8n 16p 2 × 8o
16p 16p 2 × 16p
[2] P42/mcm a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2b 2a 2d 2c 4f 4e 4g 4h 8k 8o 4i; 4j
(132) 8l 8n 8m 8n 16p 2 × 8n 2 × 8o
16p 16p 2 × 16p
[2] P42/mmc a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2a 2e 2b 2f 4i 2c; 2d 4g 4h 2 × 4i 8q 8n
(131) 4j; 4l 8o 4k; 4m 8p 2 × 8q 16r 16r
2 × 8o 2 × 8p 2 × 16r
[2] P42/mmc a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2e 2a 2f 2b 2c; 2d 4i 4g 4h 2 × 4i 8n 8q
(131) 8o 4j; 4l 8p 4k; 4m 16r 2 × 8q 16r
2 × 8o 2 × 8p 2 × 16r
[2] P4/nmm a - b, a + b, c origin 1 : (1/2)(x - y), 2a 2b 2c 2c 4e 4d 4f 2 × 2c 8j 8i 8i
(129) (1/2)(x + y), z; +((1/2), (1/2), 0) 8g 8h 8j 8j 16k 16k 2 × 8i
origin 2 : (1/2)(x - y) - (1/4), 16k 2 × 8j 2 × 16k
(1/2)(x + y) + (1/4), z; +((1/2), (1/2), 0)
[2] P4/nmm a - b, a + b, c origin 1 : (1/2)(x - y) + (1/2), 2c 2c 2a 2b 4e 4d 2 × 2c 4f 8j 8i 8i
(129) (1/2)(x + y), z; +((1/2), (1/2), 0) 8j 8j 8g 8h 16k 16k 2 × 8i
origin 2 : (1/2)(x - y) + (1/4), 2 × 8j 16k 2 × 16k
(1/2)(x + y) + (1/4), z; +((1/2), (1/2), 0)
[2] P4/mbm a - b, a + b, c (1/2)(x - y), (1/2)(x + y), z; 2a 2b 2d 2c 4h 4g 4e 4f 8k 8i 8j
(127) +((1/2), (1/2), 0) 8i 8j 2 × 4g 2 × 4h 2 × 8i 2 × 8j 16l
16l 2 × 8k 2 × 16l
[2] P4/mbm a - b, a + b, c (1/2)(x - y) + (1/2), (1/2)(x + y), z; 2d 2c 2a 2b 4h 4g 4f 4e 8k 8i 8j
(127) +((1/2), (1/2), 0) 2 × 4g 2 × 4h 8i 8j 2 × 8i 2 × 8j 16l
2 × 8k 16l 2 × 16l
[2] P4/nbm a - b, a + b, c origin 1 : (1/2)(x - y), 2a 2b 2c 2d 4f 4e 4g 4h 8m 8k 8l
(125) (1/2)(x + y), z; +((1/2), (1/2), 0) 8i 8j 8m 8m 16n 16n 16n
origin 2 : (1/2)(x - y) + (1/4), 16n 2 × 8m 2 × 16n
(1/2)(x + y) + (1/4), z; +((1/2), (1/2), 0)
[2] P4/nbm a - b, a + b, c origin 1 : (1/2)(x - y) + (1/2), 2c 2d 2a 2b 4f 4e 4h 4g 8m 8k 8l
(125) (1/2)(x + y), z; +((1/2), (1/2), 0) 8m 8m 8i 8j 16n 16n 16n
origin 2 : (1/2)(x - y) - (1/4), 2 × 8m 16n 2 × 16n
(1/2)(x + y) + (1/4), z; +((1/2), (1/2), 0)
[2] P4/mcc a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2b 2a 2d 2c 4f 4e 4g 4h 8i 8m 8j
(124) 8m 8k 8m 8l 2 × 8m 16n 16n
16n 16n 2 × 16n
[2] P4/mcc a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2a 2b 2c 2d 4e 4f 4g 4h 8i 8j 8m
(124) 8k 8m 8l 8m 16n 2 × 8m 16n
16n 16n 2 × 16n


Axes Coordinates Wyckoff positions
  1a 1b 1c 1d 2e 2f
2g 2h 4i 4j 4k 4l
4m 4n 4o 8p 8q
8r 8s 8t 16u
   Enlarged unit cell, isomorphic
[2] P4/mmm a, b, 2c x, y, (1/2)z; 1a; 1b 2g 1c; 1d 2h 4i 2e; 2f
+(0, 0, (1/2)) 2 × 2g 2 × 2h 2 × 4i 4j; 4k 8r 4l; 4m
8s 4n; 4o 8t 8p; 8q 16u
2 × 8r 2 × 8s 2 × 8t 2 × 16u
[2] P4/mmm a, b, 2c x, y, (1/2)z + (1/4); 2g 1a; 1b 2h 1c; 1d 2e; 2f 4i
+(0, 0, (1/2)) 2 × 2g 2 × 2h 2 × 4i 8r 4j; 4k 8s
4l; 4m 8t 4n; 4o 16u 8p; 8q
2 × 8r 2 × 8s 2 × 8t 2 × 16u
[3] P4/mmm a, b, 3c x, y, (1/3)z; 1a; 2g 1b; 2g 1c; 2h 1d; 2h 2e; 4i 2f; 4i
±(0, 0, (1/3)) 3 × 2g 3 × 2h 3 × 4i 4j; 8r 4k; 8r 4l; 8s
4m; 8s 4n; 8t 4o; 8t 8p; 16u 8q; 16u
3 × 8r 3 × 8s 3 × 8t 3 × 16u
[p] P4/mmm 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
+(0, 0, (u/p)) p × 2g p × 2h p × 4i 4j; ((p - 1)/2) × 8r 4k; ((p - 1)/2) × 8r 4l; ((p - 1)/2) × 8s
p = prime > 2; 4m; ((p - 1)/2) × 8s 4n; ((p - 1)/2) × 8t 4o; ((p - 1)/2) × 8t 8p; ((p - 1)/2) × 16u 8q; ((p - 1)/2) × 16u
u = 1, . . ., p - 1 p × 8r p × 8s p × 8t p × 16u
[2] P4/mmm a - b, (1/2)(x - y), 1a; 1c 1b; 1d 2f 2e 4k 4j
a + b, c (1/2)(x + y), z; 2g; 2h 4i 8r 4l; 4n 4m; 4o 2 × 4j
+((1/2), (1/2), 0) 2 × 4k 8p 8q 2 × 8p 2 × 8q
8s; 8t 2 × 8r 16u 2 × 16u
[2] P4/mmm a - b, (1/2)(x - y) + (1/2), 2f 2e 1a; 1c 1b; 1d 4k 4j
a + b, c (1/2)(x + y), z; 4i 2g; 2h 8r 4l; 4n 4m; 4o 8p
+((1/2), (1/2), 0) 8q 2 × 4j 2 × 4k 2 × 8p 2 × 8q
8s; 8t 16u 2 × 8r 2 × 16u
[9] P4/mmm 3a, 3b, c (1/3)x, (1/3)y, z; 1a; 4j; 4l 1b; 4k; 4m 1c; 4j; 4n 1d; 4k; 4o 2e; 4m; 4o; 8q 2f; 4l; 4n; 8p
±((1/3), 0, 0); 2g; 8r; 8s 2h; 8r; 8t 4i; 8s; 8t; 16u 3 × 4j; 3 × 8p 3 × 4k; 3 × 8q 3 × 4l; 3 × 8p
±(0, (1/3), 0); 3 × 4m; 3 × 8q 3 × 4n; 3 × 8p 3 × 4o; 3 × 8q 9 × 8p 9 × 8q
±((1/3), (1/3), 0); 3 × 8r; 3 × 16u 3 × 8s; 3 × 16u 3 × 8t; 3 × 16u 9 × 16u
±((1/3), (2/3), 0)
[p2] P4/mmm 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) × 4m; 2f; ((p - 1)/2) × 4l;
+((u/p), (v/p), 0) ((p - 1)/2) × 4l; ((p - 1)/2) × 4m; ((p - 1)/2) × 4n; ((p - 1)/2) × 4o; ((p - 1)/2) × 4o; ((p - 1)/2) × 4n;
p = prime > 2; (((p - 1)(p - 3))/8) × 8p (((p - 1)(p - 3))/8) × 8q (((p - 1)(p - 3))/8) × 8p (((p - 1)(p - 3))/8) × 8q (((p - 1)2)/4) × 8q (((p - 1)2)/4) × 8p
u, v = 1, . . ., p - 1 2g; ((p - 1)/2) × 8r; 2h; ((p - 1)/2) × 8r; 4i; ((p - 1)/2) × 8s; p × 4j; p × 4k; p × 4l;
((p - 1)/2) × 8s; ((p - 1)/2) × 8t; ((p - 1)/2) × 8t; ((p(p - 1))/2) × 8p ((p(p - 1))/2) × 8q ((p(p - 1))/2) × 8p
(((p - 1)(p - 3))/8) × 16u (((p - 1)(p - 3))/8) × 16u (((p - 1)2)/4) × 16u
p × 4m; p × 4n; p × 4o; p2 × 8p p2 × 8q
((p(p - 1))/2) × 8q ((p(p - 1))/2) × 8p ((p(p - 1))/2) × 8q
p × 8r; p × 8s; p × 8t; p2 × 16u
((p(p - 1))/2) × 16u ((p(p - 1))/2) × 16u ((p(p - 1))/2) × 16u









































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