Tetragonal (422) 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.

(

089-p422.gif (2706 bytes)
89   p422)

(+x,+y,+z); (-x,-y,+z); (-x,+y,-z); (+x,-y,-z)
(+y,-x,+z); (-y,+x,+z); ( +y,+x,-z); (-y,-x,-z)

090p42i2.gif (2308 bytes)
90   p42₁2)

(+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)

091p4i22.gif (2437 bytes)
91   p4₁22)

(+x,+y,+z); (-x,-y,1/2+z -y,+x,1/4+z); (+y,-x,3/4+z)
(-x,+y,-z); (+x,-y,1/2-z -y,-x,1/4-z); (+y,+x,3/4-z); ()

092p4i2i.gif (2194 bytes)
92   p4₁2₁2)

(+x,+y,+z); (-x,-y, 1/2+z);
(1/2-y, 1/2+x, 1/4+z); (1/2+y, 1/2-x,3/4+z);
(+y,+x,-z); (-y,-x, 1/2-z);
(1/2-x, 1/2+y, 1/4-z); (1/2+x, 1/2-y, 3/4-z)

093p4j22.gif (2545 bytes)
93   p42 22)

(+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)

094p4j2i.gif (2324 bytes)
94   p42 2₁2)

(+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)

095p4k22.gif (2430 bytes)
95   p4₃ 22)

(+x,+y,+z); (-x,-y,1/2+z);
(+y,-x,1/4+z); (-y,+x,3/4+z )
(-x,+y,-z); (+x,-y,1/2-z);
(+y,+x,1/4-z); (-y,-x,3/4-z)

096p4k2i.gif (2173 bytes)
96   p4₃ 2₁2)

(+x,+y,+z); (-x,-y,1/2+z);
(1/2+y, 1/2-x,1/4+z); (1/2-y, 1/2+x, 3/4+z);
(-x,+y,-z); (+x,-y,1/2-z);
(1/2+y, 1/2+x,1/4-z); (1/2-y, 1/2-x,3/4-z)

097-i422.gif (3076 bytes)
97   i422

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)

098i4i22.gif (3164 bytes)
98   i4₁22

Origins: (0,0,0); (1/2.1/2,1/2)

(+x,+y,+z); (-x,-y,+z);
(-x,1/2+y,1/4-z); (+x,1/2-y,1/4-z);
(+y,+x,-z); (-y,-x,-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 31 May 2020

(

	89 p422) (+x,+y,+z); (-x,-y,+z); (-x,+y,-z); (+x,-y,-z) (+y,-x,+z); (-y,+x,+z); ( +y,+x,-z); (-y,-x,-z)
	90 p42₁2) (+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)
	91 p4₁22) (+x,+y,+z); (-x,-y,1/2+z -y,+x,1/4+z); (+y,-x,3/4+z) (-x,+y,-z); (+x,-y,1/2-z -y,-x,1/4-z); (+y,+x,3/4-z); ()
	92 p4₁2₁2) (+x,+y,+z); (-x,-y, 1/2+z); (1/2-y, 1/2+x, 1/4+z); (1/2+y, 1/2-x,3/4+z); (+y,+x,-z); (-y,-x, 1/2-z); (1/2-x, 1/2+y, 1/4-z); (1/2+x, 1/2-y, 3/4-z)
	93 p42 22) (+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)
	94 p42 2₁2) (+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)
	95 p4₃ 22) (+x,+y,+z); (-x,-y,1/2+z); (+y,-x,1/4+z); (-y,+x,3/4+z ) (-x,+y,-z); (+x,-y,1/2-z); (+y,+x,1/4-z); (-y,-x,3/4-z)
	96 p4₃ 2₁2) (+x,+y,+z); (-x,-y,1/2+z); (1/2+y, 1/2-x,1/4+z); (1/2-y, 1/2+x, 3/4+z); (-x,+y,-z); (+x,-y,1/2-z); (1/2+y, 1/2+x,1/4-z); (1/2-y, 1/2-x,3/4-z)
	97 i422 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)
	98 i4₁22 Origins: (0,0,0); (1/2.1/2,1/2) (+x,+y,+z); (-x,-y,+z); (-x,1/2+y,1/4-z); (+x,1/2-y,1/4-z); (+y,+x,-z); (-y,-x,-z); (-y,1/2+x,1/4+z); (+y,1/2-x,1/4+z)