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Result of symbol lookup:
  Space group number: 213
  Schoenflies symbol: O^7
  Hermann-Mauguin symbol: P 41 3 2
  Hall symbol: P 4bd 2ab 3

Input space group symbol: P 41 3 2
Convention: Default

Number of lattice translations: 1
Space group is acentric.
Space group is chiral.
Space group is enantiomorphic.
Number of representative symmetry operations: 24
Total number of symmetry operations: 24

Parallelepiped containing an asymmetric unit:
  1/8<=x<=3/8; 1/8<=y<=3/8; 0<=z<1

List of symmetry operations:
Matrix Rotation-part type Axis direction Screw/glide component Origin shift
x,y,z 1---
-y+1/4,x+3/4,z+1/4 4^1[0,0,1]0,0,1/4-1/4,1/2,0
-x+1/2,-y,z+1/2 2[0,0,1]0,0,1/21/4,0,0
y+1/4,-x+1/4,z+3/4 4^-1[0,0,1]0,0,3/41/4,0,0
x+1/2,-y+1/2,-z 2[1,0,0]1/2,0,00,1/4,0
y+3/4,x+1/4,-z+1/4 2[1,1,0]1/2,1/2,01/4,0,1/8
-x,y+1/2,-z+1/2 2[0,1,0]0,1/2,00,0,1/4
-y+3/4,-x+3/4,-z+3/4 2[-1,1,0]0,0,03/4,0,3/8
z,x,y 3^1[1,1,1]0,0,00,0,0
-x+1/4,z+3/4,y+1/4 2[0,1,1]0,1/2,1/21/8,1/4,0
-z+1/2,-x,y+1/2 3^-1[-1,1,1]0,0,01/2,-1/2,0
x+1/4,-z+1/4,y+3/4 4^1[1,0,0]1/4,0,00,-1/4,1/2
z+1/2,-x+1/2,-y 3^-1[1,-1,1]0,0,01/2,0,0
x+3/4,z+1/4,-y+1/4 4^-1[1,0,0]3/4,0,00,1/4,0
-z,x+1/2,-y+1/2 3^1[-1,-1,1]0,0,00,1/2,0
-x+3/4,-z+3/4,-y+3/4 2[0,-1,1]0,0,03/8,3/4,0
y,z,x 3^-1[1,1,1]0,0,00,0,0
y+1/2,-z+1/2,-x 3^-1[-1,-1,1]1/3,1/3,-1/31/3,1/6,0
z+3/4,y+1/4,-x+1/4 4^1[0,1,0]0,1/4,01/2,0,-1/4
-y,z+1/2,-x+1/2 3^1[-1,1,1]-1/3,1/3,1/31/6,1/6,0
-z+3/4,-y+3/4,-x+3/4 2[-1,0,1]0,0,03/4,3/8,0
-y+1/2,-z,x+1/2 3^1[1,-1,1]1/3,-1/3,1/3-1/6,1/3,0
z+1/4,-y+1/4,x+3/4 2[1,0,1]1/2,0,1/2-1/4,1/8,0
-z+1/4,y+3/4,x+1/4 4^-1[0,1,0]0,3/4,00,0,1/4 

List of Wyckoff positions:
Wyckoff letter Multiplicity Site symmetry
point group type 		Representative special position operator
e241x,y,z
d1221/8,y,y+1/4
c83x,x,x
b432-1/8,-1/8,-1/8
a4323/8,3/8,3/8 

Harker planes:
Algebraic Normal vector A point in the plane
x-y+1/4,-x-y+3/4,1/4[0,0,1]1/4,3/4,1/4
2*x+1/2,2*y,1/2[0,0,1]1/2,0,1/2
1/2,2*y+1/2,2*z[1,0,0]1/2,1/2,0
x+y+3/4,-x-y+1/4,2*z+1/4[1,1,0]3/4,1/4,1/4
2*x,1/2,2*z+1/2[0,1,0]0,1/2,1/2
x+y+3/4,x+y+3/4,2*z+3/4[-1,1,0]3/4,3/4,3/4
x+z,-x-y,y-z[1,1,1]0,0,0
2*x+1/4,y+z+3/4,-y-z+1/4[0,1,1]1/4,3/4,1/4
1/4,y-z+1/4,-y-z+3/4[1,0,0]1/4,1/4,3/4
x+z,-x-y+1/2,-y+z+1/2[-1,-1,1]0,1/2,1/2
2*x+3/4,y+z+3/4,y+z+3/4[0,-1,1]3/4,3/4,3/4
x-z+3/4,1/4,x+z+1/4[0,1,0]3/4,1/4,1/4
x+y,y-z+1/2,x+z+1/2[-1,1,1]0,1/2,1/2
x+z+3/4,2*y+3/4,x+z+3/4[-1,0,1]3/4,3/4,3/4
x+y+1/2,y-z,-x-z+1/2[1,-1,1]1/2,0,1/2
x+z+1/4,2*y+1/4,-x-z+3/4[1,0,1]1/4,1/4,3/4 

Additional generators of Euclidean normalizer:
  Number of structure-seminvariant vectors and moduli: 1
    Vector    Modulus
    (1, 1, 1) 2

Grid factors implied by symmetries:
  Space group: (4, 4, 4)
  Structure-seminvariant vectors and moduli: (2, 2, 2)
  Euclidean normalizer: (4, 4, 4)

  All points of a grid over the unit cell are mapped
  exactly onto other grid points only if the factors
  shown above are factors of the grid.
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