[script p]-31c -31m Trigonal
No. 52 [script p]-312/c Patterson symmetry [script p]-31m
FIRST SETTING

symmetry group diagram

Origin at centre (-3) at -31c

Asymmetric unit 0 ≤ x; 0 ≤ y; 0 ≤ z ≤ 1/4

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-x1/4
      (23 | 0, 0, 1/2)
(5)  2   x, 2x1/4
      (22 | 0, 0, 1/2)
(6)  2   2xx1/4
      (21 | 0, 0, 1/2)
(7)  -1   0, 0, 0
      (-1 | 0, 0, 0)
(8)  -3+   0, 0, z; 0, 0, 0
      (-3z | 0, 0, 0)
(9)  -3-   0, 0, z; 0, 0, 0
      (-3z-1 | 0, 0, 0)
(10)  c   xxz
      (m3 | 0, 0, 1/2)
(11)  c   x, 0, z
      (m2 | 0, 0, 1/2)
(12)  c   0, yz
      (m1 | 0, 0, 1/2)

Generators selected (1); t(0, 0, 1); (2); (4); (7)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

 General:
12 e 1
(1) xyz(2) -yx - yz(3) -x + y-xz
(4) -y-x-z + 1/2(5) -x + yy-z + 1/2(6) xx - y-z + 1/2
(7) -x-y-z(8) y-x + y-z(9) x - yx-z
(10) yxz + 1/2(11) x - y-yz + 1/2(12) -x-x + yz + 1/2
l: l = 2n
  Special: no extra conditions
6 d  . . 2 
x-x1/4x, 2x1/4-(2x), -x1/4-xx3/4-x, -(2x), 3/42xx3/4
4 c  3 . . 
0, 0, z0, 0, -z + 1/20, 0, -z0, 0, z + 1/2
2 b  -3 . . 
0, 0, 00, 0, 1/2
2 a  3 . 2 
0, 0, 1/40, 0, 3/4

Symmetry of special projections

Along [001]   6mm

Origin at 0, 0, z
Along [100]   [script p]2mg
a' = c   
Origin at x, 0, 0
Along [210]   [script p]211
a' = 1/2c   
Origin at x1/2x, 0

Maximal non-isotypic non-enantiomorphic subgroups


I[2] [script p]31c ([script p]3c1, 50)1; 2; 3; 10; 11; 12
 [2] [script p]312 (46)1; 2; 3; 4; 5; 6
 [2] [script p]-311 ([script p]-3, 45)1; 2; 3; 7; 8; 9
 [3] [script p]112/c ([script p]2/c11, 7)1; 4; 7; 10
 [3] [script p]112/c ([script p]2/c11, 7)1; 5; 7; 11
 [3] [script p]112/c ([script p]2/c11, 7)1; 6; 7; 12
IIa none
IIbnone

Maximal isotypic subgroups and enantiomorphic subgroups of lowest index


IIc[3] [script p]-31c (c' = 3c) (52)

Minimal non-isotypic non-enantiomorphic supergroups


I[2] [script p]6/mcc (74); [2] [script p]63/mmc (75)
II[2] [script p]-31m (c' = 1/2c) (51)
[script p]-3c1 -3m1 Trigonal
No. 52 [script p]-32/c1 Patterson symmetry [script p]-3m1
SECOND SETTING

symmetry group diagram

Origin at centre (-3) at -3c1

Asymmetric unit 0 ≤ x; 0 ≤ y; 0 ≤ z ≤ 1/4

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   xx1/4
      (2xy | 0, 0, 1/2)
(5)  2   x, 0, 1/4
      (2x | 0, 0, 1/2)
(6)  2   0, y1/4
      (2y | 0, 0, 1/2)
(7)  -1   0, 0, 0
      (-1 | 0, 0, 0)
(8)  -3+   0, 0, z; 0, 0, 0
      (-3z | 0, 0, 0)
(9)  -3-   0, 0, z; 0, 0, 0
      (-3z-1 | 0, 0, 0)
(10)  c   x-xz
      (mxy | 0, 0, 1/2)
(11)  c   x, 2xz
      (mx | 0, 0, 1/2)
(12)  c   2xxz
      (my | 0, 0, 1/2)

Generators selected (1); t(0, 0, 1); (2); (4); (7)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

 General:
12 e 1
(1) xyz(2) -yx - yz(3) -x + y-xz
(4) yx-z + 1/2(5) x - y-y-z + 1/2(6) -x-x + y-z + 1/2
(7) -x-y-z(8) y-x + y-z(9) x - yx-z
(10) -y-xz + 1/2(11) -x + yyz + 1/2(12) xx - yz + 1/2
l: l = 2n
  Special: no extra conditions
6 d  . 2 . 
x, 0, 1/40, x1/4-x-x1/4-x, 0, 3/40, -x3/4xx3/4
4 c  3 . . 
0, 0, z0, 0, -z + 1/20, 0, -z0, 0, z + 1/2
2 b  -3 . . 
0, 0, 00, 0, 1/2
2 a  3 2 . 
0, 0, 1/40, 0, 3/4

Symmetry of special projections

Along [001]   6mm

Origin at 0, 0, z
Along [100]   [script p]211
a' = 1/2c   
Origin at x, 0, 0
Along [210]   [script p]2mg
a' = c   
Origin at x1/2x, 0

Maximal non-isotypic non-enantiomorphic subgroups


I[2] [script p]3c1 (50)1; 2; 3; 10; 11; 12
 [2] [script p]321 ([script p]312, 46)1; 2; 3; 4; 5; 6
 [2] [script p]-311 ([script p]-3, 45)1; 2; 3; 7; 8; 9
 [3] [script p]12/c1 ([script p]2/c11, 7)1; 4; 7; 10
 [3] [script p]12/c1 ([script p]2/c11, 7)1; 5; 7; 11
 [3] [script p]12/c1 ([script p]2/c11, 7)1; 6; 7; 12
IIa none
IIbnone

Maximal isotypic subgroups and enantiomorphic subgroups of lowest index


IIc[3] [script p]-3c1 (c' = 3c) ([script p]-31c, 52)

Minimal non-isotypic non-enantiomorphic supergroups


I[2] [script p]6/mcc (74); [2] [script p]63/mmc (75)
II[2] [script p]-31m (c' = 1/2c) (51)








































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