Hexagonal (6/m 2/m 2/m) 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

191-p6mm.gif (7026 bytes) 191     P6/mmm

(+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);
(+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);

192-p6mc.gif (6668 bytes) 192     P6/mcc

(+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);
(+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);

193-p6km.gif (6813 bytes) 193      P63/mcm

(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z);
(-x, -y, 1/2+z); (+y, +y-x, 1/2+z); (+x-y, +x, 1/2+z);
(+y, +x, +z); (-x, +y-x, +z); (+x-y, -y, +z);
(-y, -x, 1/2+z); (+x, +x-y, 1/2+z); (+y-x, +y, 1/2+z);
(+x, +y, 1/2-z); (-y, +x-y, 1/2-z); (+y-x, -x, 1/2-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, -z); (+x, +x-y, -z); (+y-x, +y, -z);

194-p6kc.gif (6788 bytes) 194      P63/mmc

(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +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);
(-y, -x, +z); (+x, +x-y, +z); (+y-x, +y, +z);
(+x, +y, 1/2-z); (-y, +x-y, 1/2-z); (+y-x, -x, 1/2-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, 1/2-z); (+x, +x-y, 1/2-z); (+y-x, +y, 1/2-z);

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Created 13 October 1999, Last Update 11 June 2020