# Tetragonal (4mm); 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.

 99      p4mm (+x, +y, +z); (-x, -y, +z); (-x, +y, +z); (+x, -y, +z); (+y, +x, +z); (-y, -x, +z); (-y, +x, +z); (+y, -x, +z); 100      p4bm (+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); 101     p42cm (+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); 102     p42nm (+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); 103     p4cc (+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) 104     p4nc (+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) 105     p42mc (+x, +y, +z); (-x, -y, +z); (-x, +y, +z); (+x, -y, +z); (+y, +x, 1/2+z); (-y, -x, 1/2+z); (-y, +x, 1/2+z); (+y, -x, 1/2+z); 106     p42bc (+x, +y, +z); (-y, +x, 1/2+z); (1/2-x, 1/2+y, +z); (1/2+y, 1/2+x, 1/2+z)(-y, -x, +z); (+y, -x, 1/2+z); (1/2+x, 1/2-y, +z); (1/2-x, 1/2-y, 1/2+z); 107     i4mm Origins: (0, 0, 0 (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, +z); (-x, +y, +z); (+x, -y, +z);(+y, +x, +z); (-y, -x, +z); (-y, +x, +z); (+y, -x, +z) 108     i4cm Origins: (0, 0, 0 (1/2.1/2, 1/2); (+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) 109     i41md Origins: (0, 0, 0 (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, +z); (+y, 1/2+x, 1/4+z); (-y, 1/2-x, 1/4+z) (-x, +y, +z); (+x, -y, +z); (-y, 1/2+x, 1/4+z); (+y, 1/2-x, 1/4+z) 110     i41cd Origins: (0, 0, 0);  (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, 1/2+z); (+y, 1/2+x, 3/4+z); (-y, 1/2-x, 3/4+z) (-x, +y, +z); (+x, -y, 1/2+z); (-y, 1/2+x, 1/4+z); (+y, 1/2-x, 1/4+z)

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Created 30 March 1999, Last Update 11 June 2020