P-3m1 No. 164 P-32/m1 D3d3


Axes Coordinates Wyckoff positions
1a 1b 2c 2d 3e 3f
6g 6h 6i 12j
I Maximal translationengleiche subgroups
[2] P3m1 (156) 1a 1a 2 × 1a 1b; 1c 3d 3d
6e 6e 2 × 3d 2 × 6e
[2] P321 (150) 1a 1b 2c 2d 3e 3f
2 × 3e 2 × 3f 6g 2 × 6g
[2] P-3 (147) 1a 1b 2c 2d 3e 3f
6g 6g 6g 2 × 6g
[3] C12/m1 2a + b, b, c (1/2)x, -(1/2)x + y, z 2a 2c 4i 4i 2b; 4e 2d; 4f
  (12) 4g; 8j 4h; 8j 4i; 8j 3 × 8j
 conjugate: a - b, a + b, c (1/2)(x - y), (1/2)(x + y), z
 conjugate: a + 2b, -a, c (1/2)y, -x + (1/2)y, z
II Maximal klassengleiche subgroups
   Enlarged unit cell, non-isomorphic
[2] P-3c1 a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2b 2a 4c 4d 6e 6f
  (165) 12g 2 × 6f 12g 2 × 12g
[2] P-3c1 a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2a 2b 4c 4d 6f 6e
  (165) 2 × 6f 12g 12g 2 × 12g
[3] P-31m 2a + b, (1/3)(x + y), (1/3)(-x + 2y), z; 1a; 2c 1b; 2d 2e; 4h 6k 3f; 6i 3g; 6j
  (162) -a + b, c ±((2/3), (1/3), 0) 3 × 6i 3 × 6j 6k; 12l 3 × 12l
   Enlarged unit cell, isomorphic
[2] P-3m1 a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 1a; 1b 2c 2 × 2c 2 × 2d 3e; 3f 6i
6g; 6h 12j 2 × 6i 2 × 12j
[2] P-3m1 a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2c 1a; 1b 2 × 2c 2 × 2d 6i 3e; 3f
12j 6g; 6h 2 × 6i 2 × 12j
[3] P-3m1 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 1a; 2c 1b; 2c 3 × 2c 3 × 2d 3e; 6i 3f; 6i
6g; 12j 6h; 12j 3 × 6i 3 × 12j
[p] P-3m1 a, b, pc x, y, (1/p)z; +(0, 0, (u/p)) 1a; ((p - 1)/2) × 2c 1b; ((p - 1)/2) × 2c p × 2c p × 2d 3e; ((p - 1)/2) × 6i 3f; ((p - 1)/2) × 6i
p = prime > 2; u = 1, . . ., p - 1 6g; ((p - 1)/2) × 12j 6h; ((p - 1)/2) × 12j p × 6i p × 12j
[4] P-3m1 2a, 2b, c (1/2)x, (1/2)y, z; +((1/2), 0, 0); 1a; 3e 1b; 3f 2c; 6i 2d; 6i 6g; 6i 6h; 6i
+(0, (1/2), 0); +((1/2), (1/2), 0) 2 × 6g; 12j 2 × 6h; 12j 2 × 6i; 12j 4 × 12j
[p2] P-3m1 pa, pb, c (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) 1a; ((p - 1)/2) × 6g; 1b; ((p - 1)/2) × 6h; 2c; 2d; 3e; ((p - 1)/2) × 6g; 3f; ((p - 1)/2) × 6h;
p = prime > 4; ((p - 1)/2) × 6i; ((p - 1)/2) × 6i; (p - 1) × 6i; (p - 1) × 6i; ((p - 1)/2) × 6i; ((p - 1)/2) × 6i;
u, v = 1, . . ., p - 1 (((p - 1)(p - 5))/12) × (((p - 1)(p - 5))/12) × (((p - 1)(p - 2))/6) × (((p - 1)(p - 2))/6) × (((p - 1)2)/4) × 12j (((p - 1)2)/4) × 12j
12j 12j 12j 12j
p × 6g; p × 6h; p × 6i; p2 × 12i
((p(p - 1))/2) × 12j ((p(p - 1))/2) × 12j ((p(p - 1))/2) × 12j









































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