p4mm 4mm Tetragonal/Square
No. 55 p4mm Patterson symmetry p4/mmm

symmetry group diagram

Origin on 4mm

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

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)  m   x, 0, z
      (my | 0, 0, 0)
(6)  m   0, yz
      (mx | 0, 0, 0)
(7)  m   x-xz
      (mxy | 0, 0, 0)
(8)  m   xxz
      (m-xy | 0, 0, 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 g 1
(1) xyz (2) -x-yz (3) -yxz (4) y-xz
(5) x-yz (6) -xyz (7) -y-xz (8) yxz
no conditions
    Special:
4 f  . m . 
x1/2z -x1/2z 1/2xz 1/2-xz
no extra conditions
4 e  . m . 
x, 0, z -x, 0, z 0, xz 0, -xz
no extra conditions
4 d  . . m 
xxz -x-xz -xxz x-xz
no extra conditions
2 c  2 mm . 
1/2, 0, z 0, 1/2z
hk: h + k = 2n
1 b  4 m m 
1/21/2z
no extra conditions
1 a  4 m m 
0, 0, z
no extra conditions

Symmetry of special projections

Along [001]   p4mm
a' = a   b' = b   
Origin at 0, 0, z
Along [100]   [script p]1m1
a' = b   
Origin at x, 0, 0
Along [110]   [script p]1m1
a' = 1/2(-a + b)   
Origin at xx, 0

Maximal non-isotypic subgroups


I [2] p411 (p4, 49) 1; 2; 3; 4
  [2] p21m (cmm2, 26) 1; 2; 7; 8
  [2] p2m1 (pmm2, 23) 1; 2; 5; 6
IIa none
IIb [2] c4md (a' = 2ab' = 2b) (p4bm, 56)

Maximal isotypic subgroups of lowest index


IIc [2] c4mm (a' = 2ab' = 2b) (p4mm, 55)

Minimal non-isotypic supergroups


I [2] p4/mmm (61); [2] p4/nmm (64)
II none








































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