p31m 3m Hexagonal No. 15 p31m Patterson symmetry p6mm

Origin at 3 1 m

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 (3)  3-   0, 0 (4)  m   x, x (5)  m   x, 0 (6)  m   0, y

Generators selected (1); t(1, 0); t(0, 1); (2); (4)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

General:
 6 d 1
 (1) x, y (2) -y, x - y (3) -x + y, -x (4) y, x (5) x - y, -y (6) -x, -x + y
no conditions
Special: as above, plus
 3 c . . m
 x, 0 0, x -x, -x
no extra conditions
 2 b 3 . .
 1/3, 2/3 2/3, 1/3
no extra conditions
 1 a 3 . m
 0, 0
no extra conditions

Maximal non-isomorphic subgroups

I[2] p311 (p3, 13)1; 2; 3
 [3] p11m (cm, 5) 1; 4 [3] p11m (cm, 5) 1; 5 [3] p11m (cm, 5) 1; 6
 IIa none
 IIb [3] h31m (a' = 3a, b' = 3b) (p3m1, 14)

Maximal isomorphic subgroups of lowest index

 IIc [4] p31m (a' = 2a, b' = 2b) (15)

Minimal non-isomorphic supergroups

 I [2] p6mm (17)
 II [3] h31m (p3m1, 14)