p3m1 3m1 Trigonal/Hexagonal
No. 69 p3m1 Patterson symmetry p-3m1

symmetry group diagram

Origin on 3m1

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

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)  m   x-xz
      (mxy | 0, 0, 0)
(5)  m   x, 2xz
      (mx | 0, 0, 0)
(6)  m   2xxz
      (my | 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) -y-xz (5) -x + yyz (6) xx - yz
no conditions
    Special: no extra conditions
3 d  . m . 
x-xz x, 2xz -(2x), -xz
1 c  3 m . 
2/31/3z
1 b  3 m . 
1/32/3z
1 a  3 m . 
0, 0, z

Symmetry of special projections

Along [001]   p3m1
a' = a   b' = b   
Origin at 0, 0, z
Along [100]   [script p]111
a' = 1/2(a + 2b)   
Origin at x, 0, 0
Along [210]   [script p]1m1
a' = 1/2b   
Origin at x1/2x, 0

Maximal non-isotypic subgroups


I [2] p311 (p3, 65) 1; 2; 3
  [3] p1m1 (cm11, 13) 1; 4
  [3] p1m1 (cm11, 13) 1; 5
  [3] p1m1 (cm11, 13) 1; 6
IIa none
IIb [3] h3m1 (a' = 3ab' = 3b) (p31m, 70)

Maximal isotypic subgroups of lowest index


IIc [4] p3m1 (a' = 2ab' = 2b) (69)

Minimal non-isotypic supergroups


I [2] p-3m1 (72); [2] p6mm (77); [2] p-6m2 (78)
II [2] h3m1 (p31m, 70)








































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