P312 No. 149 D31


Axes Coordinates Wyckoff positions
1a 1b 1c 1d 1e 1f 2g 2h 2i 3j
3k 6l
I Maximal translationengleiche subgroups
[2] P3 (143) 1a 1a 1b 1b 1c 1c 2 × 1a 2 × 1b 2 × 1c 3d
3d 2 × 3d
[3] C121 a, a + 2b, c x - (1/2)y, (1/2)y, z 2a 2b 2a 2b 2a 2b 4c 4c 4c 2a; 4c
  (5) 2b; 4c 3 × 4c
 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
[3] R32 a - b, (1/3)(2x - y), (1/3)(x + y), (1/3)z; 3a; 6c 3b; 6c 9d 9e 9d 9e 3 × 6c 18f 18f 9d; 18f
 (155) a + 2b, 3c ±(0, 0, (1/3)) 9e; 18f 3 × 18f
(hexagonal axes)
[3] R32 a - b, (1/3)(2x - y) - (1/3), (1/3)(x + y), (1/3)z; 9d 9e 3a; 6c 3b; 6c 9d 9e 18f 3 × 6c 18f 9d; 18f
 (155) a+2b, 3c ±(0, 0, (1/3)) 9e; 18f 3 × 18f
(hexagonal axes)
[3] R32 a - b, (1/3)(2x - y) + (1/3), (1/3)(x + y), (1/3)z; 9d 9e 9d 9e 3a; 6c 3b; 6c 18f 18f 3 × 6c 9d; 18f
 (155) a + 2b, 3c ±(0, 0, (1/3)) 9e; 18f 3 × 18f
(hexagonal axes)
[3] R32 2a + b, (1/3)(x + y), (1/3)(-x + 2y), (1/3)z; 3a; 6c 3b; 6c 9d 9e 9d 9e 3 × 6c 18f 18f 9d; 18f
 (155) -a + b, 3c ±(0, 0, (1/3)) 9e; 18f 3 × 18f
(hexagonal axes)
[3] R32 2a + b, (1/3)(x + y) - (1/3), (1/3)(-x + 2y), (1/3)z; 9d 9e 9d 9e 3a; 6c 3b; 6c 18f 18f 3 × 6c 9d; 18f
 (155) -a + b, 3c ±(0, 0, (1/3)) 9e; 18f 3 × 18f
(hexagonal axes)
[3] R32 2a + b, (1/3)(x + y) + (1/3), (1/3)(-x + 2y), (1/3)z; 9d 9e 3a; 6c 3b; 6c 9d 9e 18f 3 × 6c 18f 9d; 18f
 (155) -a + b, 3c ±(0, 0, (1/3)) 9e; 18f 3 × 18f
(hexagonal axes)
[3] P3212 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 3a 3b 3a 3b 3a 3b 6c 6c 6c 3a; 6c
 (153) 3b; 6c 3 × 6c
[3] P3112 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 3a 3b 3a 3b 3a 3b 6c 6c 6c 3a; 6c
 (151) 3b; 6c 3 × 6c
[3] P321 2a + b, (1/3)(x + y), (1/3)(-x + 2y), z; 1a; 2d 1b; 2d 3e 3f 3e 3f 2c; 2 × 2d 6g 6g 3e; 6g
 (150) -a + b, c ±((1/3), (2/3), 0) 3f; 6g 3 × 6g
[3] P321 2a + b, (1/3)(x + y) - (1/3), (1/3)(-x + 2y), z; 3e 3f 3e 3f 1a; 2d 1b; 2d 6g 6g 2c; 2 × 2d 3e; 6g
 (150) -a + b, c ±((1/3), (2/3), 0) 3f; 6g 3 × 6g
[3] P321 2a + b, (1/3)(x + y) + (1/3), (1/3)(-x + 2y), z; 3e 3f 1a; 2d 1b; 2d 3e 3f 6g 2c; 2 × 2d 6g 3e; 6g
 (150) -a + b, c ±((1/3), (2/3), 0) 3f; 6g 3 × 6g
   Enlarged unit cell, isomorphic
[2] P312 a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 1a; 1b 2g 1c; 1d 2h 1e; 1f 2i 2 × 2g 2 × 2h 2 × 2i 3j; 3k
6l 2 × 6l
[2] P312 a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 2g 1a; 1b 2h 1c; 1d 2i 1e; 1f 2 × 2g 2 × 2h 2 × 2i 6l
3j; 3k 2 × 6l
[3] P312 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 1a; 2g 1b; 2g 1c; 2h 1d; 2h 1e; 2i 1f; 2i 3 × 2g 3 × 2h 3 × 2i 3j; 6l
3k; 6l 3 × 6l


Axes Coordinates Wyckoff positions
1a 1b 1c 1d 1e 1f 2g
2h 2i 3j 3k 6l
[p] P312 a, b, pc x, y, (1/p)z; +(0, 0, (u/p)) 1a; ((p - 1)/2) × 2g 1b; ((p - 1)/2) × 2g 1c; ((p - 1)/2) × 2h 1d; ((p - 1)/2) × 2h 1e; ((p - 1)/2) × 2i 1f; ((p - 1)/2) × 2i p × 2g
p = prime > 2; u = 1, . . ., p - 1 p × 2h p × 2i 3j; ((p - 1)/2) × 6l 3k; ((p - 1)/2) × 6l p × 6l
[4] P312 2a, 2b, c (1/2)x, (1/2)y, z; +((1/2), 0, 0); 1a; 3j 1b; 3k 1e; 3j 1f; 3k 1c; 3j 1d; 3k 2g; 6l
+(0, (1/2), 0); +((1/2), (1/2), 0) 2i; 6l 2h; 6l 2 × 3j; 6l 2 × 3k; 6l 4 × 6l
[p2] P312 pa, pb, c (1/p)x, (1/p)y, z; +((u/p), (v/p), 0) 1a; 1b; 1c(e*); 1d(f*); 1e(c*); 1f(d*); 2g; ((p2 - 1)/3) × 6l
p = prime > 4; (p - 1) × 3j; (p - 1) × 3k; (p - 1) × 3j; (p - 1) × 3k; (p - 1) × 3j; (p - 1) × 3k;
u, v = 1, . . ., p - 1 (((p - 1)(p - 2))/6) × 6l (((p - 1)(p - 2))/6) × 6l (((p - 1)(p - 2))/6) × 6l (((p - 1)(p - 2))/6) × 6l (((p - 1)(p - 2))/6) × 6l (((p - 1)(p - 2))/6) × 6l
2h(i*); 2i(h*); p × 3j; p × 3k; p2 × 6l
((p2 - 1)/3) × 6l ((p2 - 1)/3) × 6l ((p(p - 1))/2) × 6l ((p(p - 1))/2) × 6l
* p = 6n - 1









































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