# Tetragonal (p4/mmm) Space Groups

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

To the right of each space group is a listing of coordinate points. These are thecoordinates to which a general point (x,y,z) is transformed by the space group. Origins(called "equivalent points" in the International Tables), areadditional points around which the points are transformed. For example, (0,0,0) refers toa corner of the unit cell, (1/2,1/2,1/2) to the center. All space groups have origin(0,0,0). For a space group with an additional origin (1/2,1/2,1/2), point (x,y,z) is alsotransformed to (1/2+x,1/2+y,1/2+z) and so on. 123    p4/mmm(+x, +y, +z); (-x ,-y, +z); (+x, +y, -z); (-x, -y, -z);(-x, +y, +z); (+x, -y, +z); (-x, +y, -z); (+x, -y, -z);(-y, +x, +z); (+y, -x, +z); (-y, +x, -z); (+y, -x, -z);(+y, +x, +z); (-y, -x, +z); (+y, +x, -z); (-y, -x, -z); 124    p4/mcc(+x, +y, +z); (-x,-y,+z); (+x, +y, -z); (-x, -y, -z);(-y, +x, +z); (+y,-x,+z); (-y, +x, -z); (+y, -x, -z);(-x, +y, 1/2+z); (+x,-y, 1/2+z); (-x, +y, 1/2-z); (+x, -y, 1/2-z)(+y, +x, 1/2+z); (-y,-x, 1/2+z); (-y, -x, 1/2-z); (+y, +x, 1/2-z) 125    p4/nbm(+x, +y, +z); (-x, -y, +z); (1/2+x, 1/2+y, -z); (1/2-x, 1/2-y, -z);(-x, +y, -z); (+x, -y, -z); (1/2+x, 1/2-y, +z); (1/2-x, 1/2+y, +z);(-y, +x, +z); (+y, -x, +z); (1/2-y, 1/2+x, -z); (1/2+y, 1/2-x, -z);(+y, +x, -z); (-y, -x, -z); (1/2+y, 1/2+x, +z); (1/2-y, 1/2-x, +z); 126    p4/ncc(+x, +y, +z); (-x, -y, +z); (1/2+x, 1/2+y, 1/2-z); (1/2-x, 1/2-y, 1/2-z);(-x, +y, -z); (+x, -y, -z); (1/2-x, 1/2+y, 1/2+z); (1/2+x, 1/2-y, 1/2+z);(-y, +x, +z); (+y, -x, +z); (1/2-y, 1/2+x, 1/2-z); (1/2+y, 1/2-x, 1/2-z); (+y, +x, -z); (-y, -x, -z); (1/2+y, 1/2+x, 1/2+z); (1/2-y, 1/2-x, 1/2+z); 127    p4/mbm(+x, +y, +z); (-x, -y, +z); (1/2+x, 1/2-y, -z); (1/2-x, 1/2+y, +z);(+x, +y, -z); (-x, +y, -z); (1/2+x, 1/2-y, -z); (1/2-x, 1/2+y, -z);(-y, +x, +z); (+y, -x, +z); (1/2+y, 1/2+x, +z); (1/2-y, 1/2-x, +z);(-y, +x, -z); (+y, -x, -z); (1/2+y, 1/2+x, -z); (1/2-y, 1/2-x, -z); 128    p4/mnc Fix figure(+x, +y, +z); (-x, -y, +z); (1/2+x, 1/2-y, 1/2-z); (1/2-x, 1/2+y, 1/2+z);(+x, +y, -z); (-x, -y, -z); (1/2+x, 1/2-y, 1/2-z); (1/2-x, 1/2+y, 1/2-z); (-y, +x, +z); (+y, -x, +z); (1/2+y, 1/2+x, 1/2+z); (1/2-y, 1/2-x, 1/2+z);(-y, +x, -z); (+y, -x, -z); (1/2+y, 1/2+x, 1/2-z); (1/2-y, 1/2-x, 1/2-z); 129    p4/nmm(+x, +y, +z); (-x, -y, +z); (1/2+x, 1/2+y, -z); (1/2-x, 1/2-y, -z);(-x, +y, +z); (+x, -y, +z); (1/2-x, 1/2+y, -z); (1/2+x, 1/2-y, -z);(-y, +x, -z); (+y, -x, -z); (1/2-y, 1/2+x, +z); (1/2+y, 1/2-x, +z);(+y, +x, -z); (-y, -x, -z); (1/2+y, 1/2+x, +z); (1/2-y, 1/2-x, +z); 130    p4/ncc(+x, +y, +z); (-x, +y, 1/2+z); (1/2+x, 1/2+y, -z); (1/2-x, 1/2+y, 1/2-z);(-x, -y, +z); (+x, -y, 1/2+z); (1/2-x, 1/2-y, -z); (1/2+x, 1/2-y, 1/2-z);(-y, +x, -z); (+y, +x, 1/2-z); (1/2-y, 1/2+x, +z); (1/2-y, 1/2-x, 1/2+z); (+y, -x, -z); (-y, -x, 1/2-z); (1/2+y, 1/2-x, +z); (1/2+y, 1/2+x, 1/2+z);