p31m 31m Trigonal/Hexagonal
No. 70 p31m Patterson symmetry p-31m

symmetry group diagram

Origin on 31m

Asymmetric unit 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 1/2; x ≤ (1 + y)/2; y ≤ min(1 - xx)
Vertices
0, 0   1/2, 0   2/31/3   1/21/2  

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)  m   xxz
      (m3 | 0, 0, 0)
(5)  m   x, 0, z
      (m2 | 0, 0, 0)
(6)  m   0, yz
      (m1 | 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 d 1
(1) xyz (2) -yx - yz (3) -x + y-xz
(4) yxz (5) x - y-yz (6) -x-x + yz
no conditions
    Special: no extra conditions
3 c  . . m 
x, 0, z 0, xz -x-xz
2 b  3 . . 
1/32/3z 2/31/3z
1 a  3 . m 
0, 0, z

Symmetry of special projections

Along [001]   p31m
a' = a   b' = b   
Origin at 0, 0, z
Along [100]   [script p]1m1
a' = 1/2(a + 2b)   
Origin at x, 0, 0
Along [210]   [script p]111
a' = 1/2b   
Origin at x1/2x, 0

Maximal non-isotypic subgroups


I [2] p311 (p3, 65) 1; 2; 3
  [3] p11m (cm11, 13) 1; 4
  [3] p11m (cm11, 13) 1; 5
  [3] p11m (cm11, 13) 1; 6
IIa none
IIb [3] h31m (a' = 3ab' = 3b) (p3m1, 69)

Maximal isotypic subgroups of lowest index


IIc [4] p31m (a' = 2ab' = 2b) (70)

Minimal non-isotypic supergroups


I [2] p-31m (71); [2] p6mm (77); [2] p-62m (79)
II [2] h31m (p3m1, 69)








































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