p321 321 Trigonal/Hexagonal
No. 68 p321 Patterson symmetry p-3m1

symmetry group diagram

Origin at 321

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   xx, 0
      (2xy | 0, 0, 0)
(5)  2   x, 0, 0
      (2x | 0, 0, 0)
(6)  2   0, y, 0
      (2y | 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) yx-z (5) x - y-y-z (6) -x-x + y-z
no conditions
    Special: no extra conditions
3 d  . 2 . 
x, 0, 0 0, x, 0 -x-x, 0
2 c  3 . . 
1/32/3z 2/31/3-z
2 b  3 . . 
0, 0, z 0, 0, -z
1 a  3 2 . 
0, 0, 0

Symmetry of special projections

Along [001]   p31m
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]11m
a' = 1/2b   
Origin at x1/2x, 0

Maximal non-isotypic subgroups


I [2] p311 (p3, 65) 1; 2; 3
  [3] p121 (c211, 10) 1; 4
  [3] p121 (c211, 10) 1; 5
  [3] p121 (c211, 10) 1; 6
IIa none
IIb [3] h321 (a' = 3ab' = 3b) (p312, 67)

Maximal isotypic subgroups of lowest index


IIc [4] p321 (a' = 2ab' = 2b) (68)

Minimal non-isotypic supergroups


I [2] p-3m1 (72); [2] p622 (76); [2] p-62m (79)
II [2] h321 (p312, 67)








































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