# Hexagonal (6mm and 6*m2) 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.

3-fold and 6-fold coordinates are tabulated with respect to axes intersecting at 60 degrees. In this oblique coordinate system, coordinates tend to be simple 183     P6mm(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (-x, -y, +z); (+y, +y-x, +z); (+x-y, +x, +z); (+y, +x, +z); (-x, +y-x, +z); (+x-y, -y, +z); (-y, -x, +z); (+x, +x-y, +z); (+y-x, +y, +z); 184     P6cc(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (-x, -y, +z); (+y, +y-x, +z); (+x-y, +x, +z); (+y, +x, 1/2+z); (-x, +y-x, 1/2+z); (+x-y, -y, 1/2+z); (-y, -x, 1/2+z); (+x, +x-y, 1/2+z); (+y-x, +y, 1/2+z); 185      P63cm(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (+y, +x, +z); (-x, +y-x, +z); (+x-y, -y, +z); (-x, -y, 1/2+z); (+y, +y-x, 1/2+z); (+x-y, +x, 1/2+z);(-y, -x, 1/2+z); (+x, +x-y, 1/2+z); (+y-x, +y, 1/2+z); 186      P63mc(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (-y, -x, +z); (+x, +x-y, +z); (+y-x, +y, +z); (-x, -y, 1/2+z); (+y, +y-x, 1/2+z); (+x-y, +x, 1/2+z); (+y, +x, 1/2+z); (-x, +y-x, 1/2+z); (+x-y, -y, 1/2+z) 187      P6*m2(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (+x, +y, -z); (-y, +x-y, -z); (+y-x, -x, -z); (-y, -x, +z); (+x, +x-y, +z); (+y-x, +y, +z); (-y, -x, -z); (+x, +x-y, -z); (+y-x, +y, -z); 188      P6*c2(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (+x, +y, 1/2-z); (-y, +x-y, 1/2-z); (+y-x, -x, 1/2-z); (-y, -x, 1/2+z); (+x, +x-y, 1/2+z); (+y-x, +y, 1/2+z); (-y, -x, -z); (+x, +x-y, -z); (+y-x, +y, -z); 189      P6*2m(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (+x, +y, -z); (-y, +x-y, -z); (+y-x, -x, -z); (+y, +x, +z); (-x, +y-x, +z); (+x-y, -y, +z); (+y, +x, -z); (-x, +y-x, -z); (+x-y, -y, -z); 189      P6*2c(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (+x, +y, 1/2-z); (-y, +x-y, 1/2-z); (+y-x, -x, 1/2-z); (+y, +x, 1/2+z); (-x, +y-x, 1/2+z); (+x-y, -y, 1/2+z); (+y, +x, -z); (-x, +y-x, -z); (+x-y, -y, -z);