R3 C34 3 Trigonal info
No. 146 R3 Patterson symmetry R-3
RHOMBOHEDRAL AXES

symmetry group diagram

Origin on 3

Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; z ≤ min(x, y)
Vertices
0, 0, 0  1, 0, 0  1, 1, 0  0, 1, 0  1, 1, 1  

Symmetry operations

(1)  1   (2)  3+   x, x, x(3)  3-   x, x, x

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry		 Coordinates Reflection conditions
 General:
3 b 1
(1) x, y, z(2) z, x, y(3) y, z, x
no conditions
    Special: as above, plus
1 a  3 . 
x, x, x
no extra conditions

Symmetry of special projections

Along [111]   p3
a' = 1/3(2a - b - c)   b' = 1/3(-a + 2b - c)   
Origin at x, x, x		Along [1-10]   p1
a' = 1/2(a + b - 2c)   b' = c   
Origin at x, -x, 0		Along [2-1-1]   p1
a' = 1/2(b - c)   b' = 1/3(a + b + c)   
Origin at 2x, -x, -x

Maximal non-isomorphic subgroups

I [3] R1 (P1, 1)1
IIa none
IIb[3] P32 (a' = a - b, b' = b - c, c' = a + b + c) (145); [3] P31 (a' = a - b, b' = b - c, c' = a + b + c) (144); [3] P3 (a' = a - b, b' = b - c, c' = a + b + c) (143)

Maximal isomorphic subgroups of lowest index

IIc[2] R3 (a' = b + c, b' = a + c, c' = a + b) (146); [4] R3 (a' = -a + b + c, b' = a - b + c, c' = a + b - c) (146)

Minimal non-isomorphic supergroups

I[2] R-3 (148); [2] R32 (155); [2] R3m (160); [2] R3c (161); [4] P23 (195); [4] F23 (196); [4] I23 (197); [4] P213 (198); [4] I213 (199)
II[3] P3 (a' = 1/3(2a - b - c), b' = 1/3(-a + 2b - c), c' = 1/3(a + b + c)) (143)








































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