p4/n 4/m Tetragonal/Square
No. 52 p4/n Patterson symmetry p4/m
ORIGIN CHOICE 1

symmetry group diagram

Origin at 4 on n at -1/4, -1/4, 0 from -1

Asymmetric unit 0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/2; 0 ≤ z

Symmetry operations

(1)  1
      (1 | 0, 0, 0)
(2)  2   0, 0, z
      (2z | 0, 0, 0)
(3)  4+   0, 0, z
      (4z | 0, 0, 0)
(4)  4-   0, 0, z
      (4z-1 | 0, 0, 0)
(5)  -1   1/41/4, 0
      (-1 | 1/21/2, 0)
(6)  n (1/21/2, 0)   xy, 0
      (mz | 1/21/2, 0)
(7)  -4+   1/2, 0, z; 1/2, 0, 0
      (-4z | 1/21/2, 0)
(8)  -4-   1/2, 0, z; 1/2, 0, 0
      (-4z-1 | 1/21/2, 0)

Generators selected (1); t(1, 0, 0); t(0, 1, 0); (2); (3); (5)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

  General:
8 e 1
(1) xyz (2) -x-yz (3) -yxz (4) y-xz
(5) -x + 1/2-y + 1/2-z (6) x + 1/2y + 1/2-z (7) y + 1/2-x + 1/2-z (8) -y + 1/2x + 1/2-z
hk: h + k = 2n
h0: h = 2n
0k: k = 2n
    Special: as above, plus
4 d  2 . . 
1/2, 0, z 0, 1/2z 0, 1/2-z 1/2, 0, -z
no extra conditions
4 c  -1 
1/41/4, 0 3/43/4, 0 3/41/4, 0 1/43/4, 0
hk: hk = 2n
2 b  4 . . 
1/21/2z 0, 0, -z
no extra conditions
2 a  -4 . . 
1/2, 0, 0 0, 1/2, 0
no extra conditions

Symmetry of special projections

Along [001]   p4
a' = 1/2(a - b)   b' = 1/2(a + b)   
Origin at 0, 0, z
Along [100]   [script p]2mg
a' = b   
Origin at x1/4, 0
Along [110]   [script p]2mm
a' = 1/2(-a + b)   
Origin at xx, 0

Maximal non-isotypic subgroups


I [2] p-4 (50) 1; 2; 7; 8
  [2] p4 (49) 1; 2; 3; 4
  [2] p2/n11 (p112/a, 7) 1; 2; 5; 6
IIa none
IIb none

Maximal isotypic subgroups of lowest index


IIc [5] p4/n (a' = a + 2b, b' = -2a + b or a' = a - 2b, b' = 2a + b) (52)

Minimal non-isotypic supergroups


I [2] p4/nbm (62); [2] p4/nmm (64)
II [2] c4/m (p4/m, 51)
p4/n  (1/4, 1/4, 0) 4/m Tetragonal/Square
No. 52 p4/n Patterson symmetry p4/m
ORIGIN CHOICE 2

symmetry group diagram

Origin at -1 on n at 1/41/4, 0 from 4

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

Symmetry operations

(1)  1
      (1 | 0, 0, 0)
(2)  2   1/41/4z
      (2z | 1/21/2, 0)
(3)  4+   1/41/4z
      (4z | 1/2, 0, 0)
(4)  4-   1/41/4z
      (4z-1 | 0, 1/2, 0)
(5)  -1   0, 0, 0
      (-1 | 0, 0, 0)
(6)  n (1/21/2, 0)   xy, 0
      (mz | 1/21/2, 0)
(7)  -4+   1/4, -1/4z; 1/4, -1/4, 0
      (-4z | 1/2, 0, 0)
(8)  -4-   -1/41/4z; -1/41/4, 0
      (-4z-1 | 0, 1/2, 0)

Generators selected (1); t(1, 0, 0); t(0, 1, 0); (2); (3); (5)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

  General:
8 e 1
(1) xyz (2) -x + 1/2-y + 1/2z (3) -y + 1/2xz (4) y-x + 1/2z
(5) -x-y-z (6) x + 1/2y + 1/2-z (7) y + 1/2-x-z (8) -yx + 1/2-z
hk: h + k = 2n
h0: h = 2n
0k: k = 2n
    Special: as above, plus
4 d  2 . . 
1/43/4z 3/41/4z 3/41/4-z 1/43/4-z
no extra conditions
4 c  -1 
0, 0, 0 1/21/2, 0 1/2, 0, 0 0, 1/2, 0
hk: hk = 2n
2 b  4 . . 
1/41/4z 3/43/4-z
no extra conditions
2 a  -4 . . 
1/43/4, 0 3/41/4, 0
no extra conditions

Symmetry of special projections

Along [001]   p4
a' = 1/2(a - b)   b' = 1/2(a + b)   
Origin at 1/41/4z
Along [100]   [script p]2mg
a' = b   
Origin at x, 0, 0
Along [110]   [script p]2mm
a' = 1/2(-a + b)   
Origin at xx, 0

Maximal non-isotypic subgroups


I [2] p-4 (50) 1; 2; 7; 8
  [2] p4 (49) 1; 2; 3; 4
  [2] p2/n11 (p112/a, 7) 1; 2; 5; 6
IIa none
IIb none

Maximal isotypic subgroups of lowest index


IIc [5] p4/n (a' = a + 2b, b' = -2a + b or a' = a - 2b, b' = 2a + b) (52)

Minimal non-isotypic supergroups


I [2] p4/nbm (62); [2] p4/nmm (64)
II [2] c4/m (p4/m, 51)








































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