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Just as the non-translational symmetry elements can be combined into point group
symbols that describe the symmetry of finite groups, so the symmetry of infinite
arrays can be summarized and symbolized. The addition of translation greatly
increases the possibilities, so that instead of the 32 point groups 230
*space groups* are needed to describe the symmetries of infinite arrays. A
complete list of these, with descriptions, is given in *International Tables for
X-Ray Crystallography* (see Section 5). All that will be attempted here is to
try to give some idea of what a space group symbol means and how to interpret
it.

Typical space group symbols are: , *C*2/*m*, *Ibca*, , *Fm*3*m*,
*P*2_{1}2_{1}2_{1}. You will notice that they all begin with a capital letter.
This gives the *lattice type* , which tells you whether the unit cell is
primitive or centred. *P* means primitive, *A, B* , or *C* means
centred on the face perpendicular to the *a, b* or *c* axis,
respectively, *F* means centred on all the faces, *I* means body
centred -- centred in the middle of the cell (from the German, *innenzentriert* )
-- and *R* means rhombohedral, which is a special type of centring unique
to the trigonal system. The group of symbols that follow give you the crystal
class, and hence the system. Thus in , the symbol tells you
that the system is triclinic (see Table 1). Likewise *C*2/*m* belongs to class
2/*m*, which you can see from Table 1 is monoclinic. and *Fm*3*m* are
equally easy to assign to the trigonal and cubic systems, respectively. *Ibca*
presents slightly more of a problem; the symbols *b*, *c* and *a* refer to glide
planes, as explained in the previous section. To find the crystal class, simply
replace any translational symmetry element by the equivalent non-translational
element: this rule holds for both glide planes and screw axes. Thus *Ibca*
belongs to class *mmm* , and is orthorhombic; the last example,
*P* 2_{1}2_{1}2_{1}, belongs to class 222 and is also orthorhombic.
The symbol also gives the positions of the various symmetry elements. Just as
*mmm* implies mirror planes perpendicular to the three mutually
perpendicular axes of the orthorhombic system, so *Ibca* tells us that in
the body-centred array there are *b* glide planes perpendicular to the
*x* axis, *c* glides perpendicular to *y* and *a* glides
perpendicular to *z* .

The page of *International Tables* describing *Pnma* is reproduced in Fig.
5.1, from which it can be seen that the *total* collection of symmetry
elements include many that are not listed in the space group symbol. Only the
essential elements are given in the symbol; redundant ones are omitted, just as
they are from point group symbols.

**Copyright © 1984, 1997 International Union of
Crystallography**