[script p]63mc 6mm Hexagonal
No. 70 [script p]63mc Patterson symmetry [script p]6/mmm
FIRST SETTING

symmetry group diagram

Origin on 3m1 on 63mc

Asymmetric unit 0 ≤ x; 0 ≤ y; 0 ≤ z ≤ 1; y ≤ x/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 (1/2)   0, 0, z
      (2z | 0, 0, 1/2)
(5)  6- (1/2)   0, 0, z
      (6z-1 | 0, 0, 1/2)
(6)  6+ (1/2)   0, 0, z
      (6z | 0, 0, 1/2)
(7)  m   x-xz
      (mxy | 0, 0, 0)
(8)  m   x, 2xz
      (mx | 0, 0, 0)
(9)  m   2xxz
      (my | 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 c 1
(1) xyz(2) -yx - yz(3) -x + y-xz
(4) -x-yz + 1/2(5) y-x + yz + 1/2(6) x - yxz + 1/2
(7) -y-xz(8) -x + yyz(9) xx - yz
(10) yxz + 1/2(11) x - y-yz + 1/2(12) -x-x + yz + 1/2
l: l = 2n
  Special: no extra conditions
6 b  . m . 
x-xzx, 2xz-(2x), -xz-xxz + 1/2-x, -(2x), z + 1/22xxz + 1/2
2 a  3 m . 
0, 0, z0, 0, z + 1/2

Symmetry of special projections

Along [001]   6mm

Origin at 0, 0, z
Along [100]   [script p]11g
a' = c   
Origin at x, 0, 0
Along [210]   [script p]11m
a' = 1/2c   
Origin at x1/2x, 0

Maximal non-isotypic non-enantiomorphic subgroups


I[2] [script p]6311 ([script p]63, 56)1; 2; 3; 4; 5; 6
 [2] [script p]31c ([script p]3c1, 50)1; 2; 3; 10; 11; 12
 [2] [script p]3m1 (49)1; 2; 3; 7; 8; 9
 [3] [script p]21mc ([script p]mc21, 17)1; 4; 7; 10
 [3] [script p]21mc ([script p]mc21, 17)1; 4; 8; 11
 [3] [script p]21mc ([script p]mc21, 17)1; 4; 9; 12
IIa none
IIbnone

Maximal isotypic subgroups and enantiomorphic subgroups of lowest index


IIc[3] [script p]63mc (c' = 3c) (70)

Minimal non-isotypic non-enantiomorphic supergroups


I[2] [script p]63/mmc (75)
II[2] [script p]6mm (c' = 1/2c) (68)
[script p]63cm 6mm Hexagonal
No. 70 [script p]63cm Patterson symmetry [script p]6/mmm
SECOND SETTING

symmetry group diagram

Origin on 31m on 63cm

Asymmetric unit 0 ≤ x; 0 ≤ y; 0 ≤ z ≤ 1; y ≤ x/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 (1/2)   0, 0, z
      (2z | 0, 0, 1/2)
(5)  6- (1/2)   0, 0, z
      (6z-1 | 0, 0, 1/2)
(6)  6+ (1/2)   0, 0, z
      (6z | 0, 0, 1/2)
(7)  c   x-xz
      (mxy | 0, 0, 1/2)
(8)  c   x, 2xz
      (mx | 0, 0, 1/2)
(9)  c   2xxz
      (my | 0, 0, 1/2)
(10)  m   xxz
      (m3 | 0, 0, 0)
(11)  m   x, 0, z
      (m2 | 0, 0, 0)
(12)  m   0, yz
      (m1 | 0, 0, 0)

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

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

Symmetry of special projections

Along [001]   6mm

Origin at 0, 0, z
Along [100]   [script p]11m
a' = 1/2c   
Origin at x, 0, 0
Along [210]   [script p]11g
a' = c   
Origin at x1/2x, 0

Maximal non-isotypic non-enantiomorphic subgroups


I[2] [script p]6311 ([script p]63, 56)1; 2; 3; 4; 5; 6
 [2] [script p]3c1 (50)1; 2; 3; 7; 8; 9
 [2] [script p]31m ([script p]3m1, 49)1; 2; 3; 10; 11; 12
 [3] [script p]21cm ([script p]mc21, 17)1; 4; 7; 10
 [3] [script p]21cm ([script p]mc21, 17)1; 4; 8; 11
 [3] [script p]21cm ([script p]mc21, 17)1; 4; 9; 12
IIa none
IIbnone

Maximal isotypic subgroups and enantiomorphic subgroups of lowest index


IIc[3] [script p]63cm (c' = 3c) ([script p]63mc, 70)

Minimal non-isotypic non-enantiomorphic supergroups


I[2] [script p]63/mmc (75)
II[2] [script p]6mm (c' = 1/2c) (68)








































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