P3m1 C3v1 3m1 Trigonal info
No. 156 P3m1 Patterson symmetry P-3m1

symmetry group diagram

Origin on 3 m 1

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

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); t(0, 0, 1); (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: as above, plus
3 d  . m . 
x, -x, zx, 2x, z (-2x), -x, z
no extra conditions
1 c  3 m . 
2/3, 1/3, z
no extra conditions
1 b  3 m . 
1/3, 2/3, z
no extra conditions
1 a  3 m . 
0, 0, z
no extra conditions

Symmetry of special projections

Along [001]   p3m1
a' = a   b' = b   
Origin at 0, 0, z		Along [100]   p1
a' = 1/2(a + 2b)   b' = c   
Origin at x, 0, 0		Along [210]   p1m1
a' = 1/2b   b' = c   
Origin at x, 1/2x, 0

Maximal non-isomorphic subgroups

I [2] P311 (P3, 143)1; 2; 3
 [brace][3] P1m1 (Cm, 8)1; 4
 [3] P1m1 (Cm, 8)1; 5
 [3] P1m1 (Cm, 8)1; 6
IIa none
IIb[2] P3c1 (c' = 2c) (158); [3] H3m1 (a' = 3a, b' = 3b) (P31m, 157)

Maximal isomorphic subgroups of lowest index

IIc[2] P3m1 (c' = 2c) (156); [4] P3m1 (a' = 2a, b' = 2b) (156)

Minimal non-isomorphic supergroups

I[2] P-3m1 (164); [2] P6mm (183); [2] P63mc (186); [2] P-6m2 (187)
II[3] H3m1 (P31m, 157); [3] R3m (obverse) (160); [3] R3m (reverse) (160)








































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