# Tetragonal (p42/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.

 131    p42/mcm(+x, +y, +z); (-x, -y, +z); (-y, +x, 1/2+z); (+y, -x, 1/2+z);(-x, +y, -z); (+x, -y, -z); (+y, +x, 1/2-z); (-y, -x, 1/2-z);(-x, -y, -z); (+x, +y, -z); (+y, -x, 1/2-z); (-y, +x, 1/2-z);(+x, -y, +z); (-x, +y, +z); (-y, -x, 1/2+z); (+y, +x, 1/2+z); 132     p42/mmc(+x, +y, +z); (-x, -y, +z); (-x, +y, 1/2+z); (+x, -y, 1/2+z);(+x, +y, -z); (-x, -y, -z); (-x, +y, 1/2-z); (+x, -y, 1/2-z);(+y, +x, +z); (-y, -x, +z); (-y, +x, 1/2+z); (+y, -x, 1/2+z);(+y, +x, -z); (-y, -x, -z); (-y, +x, 1/2-z); (+y, -x, 1/2-z); 133     p42/nbc(+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); 134     p42/nnm(+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); 135     p42/mbc(+x, +y, +z); (-y, +x, 1/2+z); (1/2+x, 1/2-y, +z); (1/2+y, 1/2+x, 1/2+z); (+x, +y, -z); (-y, +x, 1/2-z); (1/2+x, 1/2-y, -z); (1/2+y, 1/2+x, 1/2-z);(-x, -y, +z); (+y, -x, 1/2+z); (1/2-x, 1/2+y, +z); (1/2-y, 1/2-x, 1/2+z);(-x, -y, -z); (+y, -x, 1/2-z); (1/2-x, 1/2+y, -z); (1/2-y, 1/2-x, 1/2-z); 136     p42/mnm(+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); 137     p42/nmc(+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); 138     p42/ncm(+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);