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

Origin on 3m1

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

Symmetry operations

 (1)  1 (2)  3+   0, 0, z (3)  3-   0, 0, z (4)  m   x, -x, z (5)  m   x, 2x, z (6)  m   2x, x, 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 e 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 d . m .
 x, -x, z x, 2x, z -(2x), -x, z
 1 c 3 m .
 2/3, 1/3, z
 1 b 3 m .
 1/3, 2/3, z
 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]   111 a' = 1/2(a + 2b)    Origin at x, 0, 0 Along [210]   1m1 a' = 1/2b    Origin at x, 1/2x, 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' = 3a, b' = 3b) (p31m, 70)

Maximal isotypic subgroups of lowest index

 IIc [4] p3m1 (a' = 2a, b' = 2b) (69)

Minimal non-isotypic supergroups

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