p321 321 Trigonal/Hexagonal No. 68 p321 Patterson symmetry p-3m1

Origin at 321

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

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)  2   x, x, 0       (2xy | 0, 0, 0) (5)  2   x, 0, 0       (2x | 0, 0, 0) (6)  2   0, y, 0       (2y | 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) 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 . 2 .
 x, 0, 0 0, x, 0 -x, -x, 0
 2 c 3 . .
 1/3, 2/3, z 2/3, 1/3, -z
 2 b 3 . .
 0, 0, z 0, 0, -z
 1 a 3 2 .
 0, 0, 0

Symmetry of special projections

 Along [001]   p31m a' = a   b' = b    Origin at 0, 0, z Along [100]   211 a' = 1/2(a + 2b)    Origin at x, 0, 0 Along [210]   11m a' = 1/2b    Origin at x, 1/2x, 0

Maximal non-isotypic subgroups

 I [2] p311 (p3, 65) 1; 2; 3 [3] p121 (c211, 10) 1; 4 [3] p121 (c211, 10) 1; 5 [3] p121 (c211, 10) 1; 6
 IIa none
 IIb [3] h321 (a' = 3a, b' = 3b) (p312, 67)

Maximal isotypic subgroups of lowest index

 IIc [4] p321 (a' = 2a, b' = 2b) (68)

Minimal non-isotypic supergroups

 I [2] p-3m1 (72); [2] p622 (76); [2] p-62m (79)
 II [2] h321 (p312, 67)