p-3 -3 Trigonal/Hexagonal
No. 66 p-3 Patterson symmetry p-3

symmetry group diagram

Origin at centre (-3)

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)  -1   0, 0, 0
      (-1 | 0, 0, 0)
(5)  -3+   0, 0, z; 0, 0, 0
      (-3z | 0, 0, 0)
(6)  -3-   0, 0, z; 0, 0, 0
      (-3z-1 | 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 e 1
(1) xyz (2) -yx - yz (3) -x + y-xz
(4) -x-y-z (5) y-x + y-z (6) x - yx-z
no conditions
    Special: no extra conditions
3 d  -1 
1/2, 0, 0 0, 1/2, 0 1/21/2, 0
2 c  3 . . 
1/32/3z 2/31/3-z
2 b  3 . . 
0, 0, z 0, 0, -z
1 a  -3 . . 
0, 0, 0

Symmetry of special projections

Along [001]   p6
a' = a   b' = b   
Origin at 0, 0, z
Along [100]   [script p]211
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] p3 (65) 1; 2; 3
  [3] p-1 (2) 1; 4
IIa none
IIb none

Maximal isotypic subgroups of lowest index


IIc [3] h-3 (a' = 3ab' = 3b) (p-3, 66)

Minimal non-isotypic supergroups


I [2] p-31m (71); [2] p-3m1 (72); [2] p6/m (75)
II none








































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