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

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 2/3, 1/3 1/2, 1/2

Symmetry operations

 (1)  1 (2)  3+   0, 0, z (3)  3-   0, 0, z (4)  m   x, x, z (5)  m   x, 0, z (6)  m   0, y, z

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) x, y, z (2) -y, x - y, z (3) -x + y, -x, z (4) y, x, z (5) x - y, -y, z (6) -x, -x + y, z
no conditions
Special: no extra conditions
 3 c . . m
 x, 0, z 0, x, z -x, -x, z
 2 b 3 . .
 1/3, 2/3, z 2/3, 1/3, z
 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]   1m1 a' = 1/2(a + 2b)    Origin at x, 0, 0 Along [210]   111 a' = 1/2b    Origin at x, 1/2x, 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' = 3a, b' = 3b) (p3m1, 69)

Maximal isotypic subgroups of lowest index

 IIc [4] p31m (a' = 2a, b' = 2b) (70)

Minimal non-isotypic supergroups

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