P-31m No. 162 P-312/m D3d1


Axes Coordinates Wyckoff positions
1a 1b 2c 2d 2e 3f
3g 4h 6i 6j 6k 12l
I Maximal translationengleiche subgroups
[2] P31m 1a 1a 2b 2b 2 × 1a 3c
  (157) 3c 2 × 2b 6d 6d 2 × 3c 2 × 6d
[2] P312 1a 1b 1c; 1e 1d; 1f 2g 3j
  (149) 3k 2h; 2i 2 × 3j 2 × 3k 6l 2 × 6l
[2] P-3 1a 1b 2d 2d 2c 3e
  (147) 3f 2 × 2d 6g 6g 6g 2 × 6g
[3] C12/m1 a, a + 2b, c x - (1/2)y, (1/2)y, z 2a 2c 4g 4h 4i 2b; 4e
  (12) 2d; 4f 8j 4g; 8j 4h; 8j 4i; 8j 3 × 8j
 conjugate: b, -2a - b, c -(1/2)x + y, -(1/2)x, z
 conjugate: -a - b, a - b, c -(1/2)(x + y), (1/2)(x - y), z
II Maximal klassengleiche subgroups
   Enlarged unit cell, non-isomorphic
[2] P-31c a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2b 2a 4f 2c; 2d 4e 6g
  (163) 6h 2 × 4f 12i 2 × 6h 12i 2 × 12i
[2] P-31c a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2a 2b 2c; 2d 4f 4e 6h
  (163) 6g 2 × 4f 2 × 6h 12i 12i 2 × 12i
[3] R-3m a - b, (1/3)(2x - y), (1/3)(x + y), (1/3)z; 3a; 6c 3b; 6c 18f 18g 3 × 6c 9e; 18h
  (166) a + 2b, 3c ±(0, 0, (1/3)) 9d; 18h 36i 18f; 36i 18g; 36i 3 × 18h 3 × 36i
(hexagonal axes)
[3] R-3m 2a + b, (1/3)(x + y), (1/3)(-x + 2y), (1/3)z; 3a; 6c 3b; 6c 18f 18g 3 × 6c 9e; 18h
  (166) -a + b, 3c ±(0, 0, (1/3)) 9d; 18h 36i 18f; 36i 18g; 36i 3 × 18h 3 × 36i
(hexagonal axes)
[3] P-3m1 a - b, (1/3)(2x - y), (1/3)(x + y), z; 1a; 2d 1b; 2d 6g 6h 2c; 2 × 2d 3e; 6i
  (164) a + 2b, c ±((2/3), (1/3), 0) 3f; 6i 12j 6g; 12j 6h; 12j 3 × 6i 3 × 12j
   Enlarged unit cell, isomorphic
[2] P-31m a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 1a; 1b 2e 2c; 2d 4h 2 × 2e 3f; 3g
6k 2 × 4h 6i; 6j 12l 2 × 6k 2 × 12l
[2] P-31m a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2e 1a; 1b 4h 2c; 2d 2 × 2e 6k
3f; 3g 2 × 4h 12l 6i; 6j 2 × 6k 2 × 12l
[3] P-31m a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 1a; 2e 1b; 2e 2c; 4h 2d; 4h 3 × 2e 3f; 6k
3g; 6k 3 × 4h 6i; 12l 6j; 12l 3 × 6k 3 × 12l
[p] P-31m a, b, pc x, y, (1/p)z; +(0, 0, (u/p)) 1a; ((p - 1)/2) × 2e 1b; ((p - 1)/2) × 2e 2c; ((p - 1)/2) × 4h 2d; ((p - 1)/2) × 4h p × 2e 3f; ((p - 1)/2) × 6k
p = prime > 2; u = 1, . . ., p - 1 3g; ((p - 1)/2) × 6k p × 4h 6i; ((p - 1)/2) × 12l 6j; ((p - 1)/2) × 12l p × 6k p × 12l
[4] P-31m 2a, 2b, c (1/2)x, (1/2)y, z; +((1/2), 0, 0); 1a; 3f 1b; 3g 2c; 6i 2d; 6j 2e; 6k 6i; 6k
+(0, (1/2), 0); +((1/2), (1/2), 0) 6j; 6k 4h; 12l 2 × 6i; 12l 2 × 6j; 12l 2 × 6k; 12l 4 × 12l
[p2] P-31m pa, pb, c (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) 1a; ((p - 1)/2) × 6i; 1b; ((p - 1)/2) × 6j; 2c; 2d; 2e; 3f; ((p - 1)/2) × 6i;
p = prime > 4; u, v = 1, . . ., p - 1 ((p - 1)/2) × 6k; ((p - 1)/2) × 6k; (p - 1) × 6i; (p - 1) × 6j; (p - 1) × 6k; ((p - 1)/2) × 6k;
(((p - 1)(p - 5))/12) × (((p - 1)(p - 5))/12) × (((p - 1)(p - 2))/6) × (((p - 1)(p - 2))/6) × (((p - 1)(p - 2))/6) × (((p - 1)2)/4) × 12l
12l 12l 12l 12l 12l
3g; ((p - 1)/2) × 6j; 4h; p × 6i; p × 6j; p × 6k; p2 × 12l
((p - 1)/2) × 6k; ((p2 - 1)/3) × 12l ((p(p - 1))/2) × 12l ((p(p - 1))/2) × 12l ((p(p - 1))/2) × 12l
(((p - 1)2)/4) × 12l









































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