p312 312 Trigonal/Hexagonal
No. 67 p312 Patterson symmetry p-31m

symmetry group diagram

Origin at 312

Asymmetric unit 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 2/3; x ≤ (1 + y)/2; y ≤ min(1 - x, (1 + x)/2); 0 ≤ z
Vertices
0, 0   1/2, 0   2/31/3   1/32/3   0, 1/2  

Symmetry operations

(1)  1
      (1 | 0, 0, 0)
(2)  3+   0, 0, z
      (3z | 0, 0, 0)
(3)  3-   0, 0, z
      (3z-1 | 0, 0, 0)
(4)  2   x-x, 0
      (23 | 0, 0, 0)
(5)  2   x, 2x, 0
      (22 | 0, 0, 0)
(6)  2   2xx, 0
      (21 | 0, 0, 0)

Generators selected (1); t(1, 0, 0); t(0, 1, 0); (2); (4)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

  General:
6 h 1
(1) xyz (2) -yx - yz (3) -x + y-xz
(4) -y-x-z (5) -x + yy-z (6) xx - y-z
no conditions
    Special: no extra conditions
3 g  . . 2 
x-x, 0 x, 2x, 0 -(2x), -x, 0
2 f  3 . . 
2/31/3z 2/31/3-z
2 e  3 . . 
1/32/3z 1/32/3-z
2 d  3 . . 
0, 0, z 0, 0, -z
1 c  3 . 2 
2/31/3, 0
1 b  3 . 2 
1/32/3, 0
1 a  3 . 2 
0, 0, 0

Symmetry of special projections

Along [001]   p3m1
a' = a   b' = b   
Origin at 0, 0, z
Along [100]   [script p]11m
a' = 1/2(a + 2b)   
Origin at x, 0, 0
Along [210]   [script p]211
a' = 1/2b   
Origin at x1/2x, 0

Maximal non-isotypic subgroups


I [2] p311 (p3, 65) 1; 2; 3
  [3] p112 (c211, 10) 1; 4
  [3] p112 (c211, 10) 1; 5
  [3] p112 (c211, 10) 1; 6
IIa none
IIb [3] h312 (a' = 3ab' = 3b) (p321, 68)

Maximal isotypic subgroups of lowest index


IIc [4] p312 (a' = 2ab' = 2b) (67)

Minimal non-isotypic supergroups


I [2] p-31m (71); [2] p622 (76); [2] p-6m2 (78)
II [2] h312 (p321, 68)








































to end of page
to top of page