p4/nmm 4/mmm Tetragonal/Square
No. 64 p4/n21/m2/m Patterson symmetry p4/mmm
ORIGIN CHOICE 1

symmetry group diagram

Origin on 4mm at -1/4, -1/4, 0 from centre (2/m)

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

Symmetry operations

(1)  1    (2)  2   0, 0, z (3)  4+   0, 0, z (4)  4-   0, 0, z
(5)  2(1/21/2, 0)   0, y, 0 (6)  2(1/21/2, 0)   x, 0, 0 (7)  2(1/21/2, 0)   xx, 0 (8)  2(1/21/2, 0)   x-x, 0
(9)  -1   1/41/4, 0 (10)  n(1/21/2, 0)   xy, 0 (11)  -4+   1/2, 0, z; 1/2, 0, 0 (12)  -4-   1/2, 0, z; 1/2, 0, 0
(13)  m   x, 0, z (14)  m   0, yz (15)  m   x-xz (16)  m   xxz

Generators selected (1); t(1, 0, 0); t(0, 1, 0); (2); (3); (5); (9)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

  General:
16 h 1
(1) xyz (2) -x-yz (3) -yxz (4) y-xz
(5) -x + 1/2y + 1/2-z (6) x + 1/2-y + 1/2-z (7) y + 1/2x + 1/2-z (8) -y + 1/2-x + 1/2-z
(9) -x + 1/2-y + 1/2-z (10) x + 1/2y + 1/2-z (11) y + 1/2-x + 1/2-z (12) -y + 1/2x + 1/2-z
(13) x-yz (14) -xyz (15) -y-xz (16) yxz
hk: h + k = 2n
0k: k = 2n
h0: h = 2n
    Special: as above, plus
8 g  . . m 
xxz -x-xz -xxz x-xz
-x + 1/2x + 1/2-z x + 1/2-x + 1/2-z x + 1/2x + 1/2-z -x + 1/2-x + 1/2-z
no extra conditions
8 f  . m . 
0, yz 0, -yz -y, 0, z y, 0, z
1/2y + 1/2-z 1/2-y + 1/2-z y + 1/21/2-z -y + 1/21/2-z
no extra conditions
8 e  . . 2 
xx + 1/2, 0 -x-x + 1/2, 0 -x + 1/2x, 0 x + 1/2-x, 0
-x + 1/2-x, 0 x + 1/2x, 0 x-x + 1/2, 0 -xx + 1/2, 0
no extra conditions
4 d  2 mm . 
1/2, 0, z 0, 1/2z 0, 1/2-z 1/2, 0, -z
no extra conditions
4 c  . . 2/m 
1/41/4, 0 3/43/4, 0 3/41/4, 0 1/43/4, 0
hk: hk = 2n
2 b  4 m m 
0, 0, z 1/21/2-z
no extra conditions
2 a  -4 m 2 
1/2, 0, 0 0, 1/2, 0
no extra conditions

Symmetry of special projections

Along [001]   p4mm
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-4m2 (59) 1; 2; 7; 8; 11; 12; 13; 14
  [2] p-421m (58) 1; 2; 5; 6; 11; 12; 15; 16
  [2] p4mm (55) 1; 2; 3; 4; 13; 14; 15; 16
  [2] p4212 (54) 1; 2; 3; 4; 5; 6; 7; 8
  [2] p4/n11 (p4/n, 52) 1; 2; 3; 4; 9; 10; 11; 12
  [2] p2/n12/m (cmme, 48) 1; 2; 7; 8; 9; 10; 15; 16
  [2] p2/n21/m1 (pmmn, 46) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none

Maximal isotypic subgroups of lowest index


IIc [9] p4/nmm (a' = 3ab' = 3b) (64)

Minimal non-isotypic supergroups


I none
II [2] c4/mmm (p4/mmm, 61)
p4/nmm  (1/4, 1/4, 0) 4/mmm Tetragonal/Square
No. 64 p4/n21/m2/m Patterson symmetry p4/mmm
ORIGIN CHOICE 2

symmetry group diagram

Origin at centre (2/m) at n21(2/m, 21/g) at 1/41/4, 0 from 4mm

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

Symmetry operations

(1)  1    (2)  2   1/41/4z (3)  4+   1/41/4z (4)  4-   1/41/4z
(5)  2(0, 1/2, 0)   0, y, 0 (6)  2(1/2, 0, 0)   x, 0, 0 (7)  2(1/21/2, 0)   xx, 0 (8)  2   x-x, 0
(9)  -1   0, 0, 0 (10)  n(1/21/2, 0)   xy, 0 (11)  -4+   1/4,-1/4z; 1/4,-1/4, 0 (12)  -4-   -1/41/4z; -1/41/4, 0
(13)  m   x1/4z (14)  m   1/4yz (15)  m   x +1/2-xz (16)  m   xxz

Generators selected (1); t(1, 0, 0); t(0, 1, 0); (2); (3); (5); (9)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions

  General:
16 h 1
(1) xyz (2) -x + 1/2-y + 1/2z (3) -y + 1/2xz (4) y-x + 1/2z
(5) -xy + 1/2-z (6) x + 1/2-y-z (7) y + 1/2x + 1/2-z (8) -y-x-z
(9) -x-y-z (10) x + 1/2y + 1/2-z (11) y + 1/2-x-z (12) -yx + 1/2-z
(13) x-y + 1/2z (14) -x + 1/2yz (15) -y + 1/2-x + 1/2z (16) yxz
hk: h + k = 2n
0k: k = 2n
h0: h = 2n
    Special: as above, plus
8 g  . . m 
xxz -x + 1/2-x + 1/2z -x + 1/2xz x-x + 1/2z
-xx + 1/2-z x + 1/2-x-z x + 1/2x + 1/2-z -x-x-z
no extra conditions
8 f  . m . 
1/4yz 1/4-y + 1/2z -y + 1/21/4z y1/4z
3/4y + 1/2-z 3/4-y-z y + 1/23/4-z -y3/4-z
no extra conditions
8 e  . . 2 
x-x, 0 -x + 1/2x + 1/2, 0 x + 1/2x, 0 -x-x + 1/2, 0
-xx, 0 x + 1/2-x + 1/2, 0 -x + 1/2-x, 0 xx + 1/2, 0
no extra conditions
4 d  2 mm . 
3/41/4z 1/43/4z 1/43/4-z 3/41/4-z
no extra conditions
4 c  . . 2/m 
0, 0, 0 1/21/2, 0 1/2, 0, 0 0, 1/2, 0
hk: hk = 2n
2 b  4 m m 
1/41/4z 3/43/4-z
no extra conditions
2 a  -4 m 2 
3/41/4, 0 1/43/4, 0
no extra conditions

Symmetry of special projections

Along [001]   p4mm
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-4m2 (59) 1; 2; 7; 8; 11; 12; 13; 14
  [2] p-421m (58) 1; 2; 5; 6; 11; 12; 15; 16
  [2] p4mm (55) 1; 2; 3; 4; 13; 14; 15; 16
  [2] p4212 (54) 1; 2; 3; 4; 5; 6; 7; 8
  [2] p4/n11 (p4/n, 52) 1; 2; 3; 4; 9; 10; 11; 12
  [2] p2/n12/m (cmme, 48) 1; 2; 7; 8; 9; 10; 15; 16
  [2] p2/n21/m1 (pmmn, 46) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none

Maximal isotypic subgroups of lowest index


IIc [9] p4/nmm (a' = 3ab' = 3b) (64)

Minimal non-isotypic supergroups


I none
II [2] c4/mmm (p4/mmm, 61)








































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