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Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be selfconsistently arranged in space. There are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Figure 4549a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. The seven holohedric point groups F^{0} and all their subgroups (merohedral groups) form the 32 crystallographic point groups.
Figure 4549a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.
Nicolaus Steno first showed in 1669 that the angles between the faces of crystals are constant, independently of the regularity of a given crystal morphology. The analysis of crystal morphologies led to the formulation of a complete set of 32 symmetry classes, called “point groups” as shown in Table 4549a. The morphologies of all crystals obey the 32 point groups. Possible symmetry elements are 1, 2, 3, 4, and 6fold rotations, mirror plane m, inversion center and a combination of rotation axis with inversion center (inversion axis). Table 4549b shows the relation between threedimensional crystal families, crystal systems, and lattice systems. 32 point groups indicate that there are only 32 possible combinations of symmetry elements. Furthermore, Table 4549c lists the symmetry groups for the crystal systems together with the point groups with Hermann–Mauguin notation.
Table 4549a. 32 point groups (Notation: Schönflies).
Table 4549b. The relation between threedimensional crystal families, crystal systems, and lattice systems.
Crystal
family 
Crystal
system 
Essential
symmetries
of point group 
Point
group 
Space
group 
Bravais
lattices 
Lattice
system 
Triclinic 
1fold axis 
2 
2 
1 
Triclinic 
Monoclinic

1 twofold axis (parallel to y) of rotation or 1 mirror plane 
3

13

2

Monoclinic 
Orthorhombic 
3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes 
3 
59 
4 
Orthorhombic 
Tetragonal 
1 fourfold axis (parallel to z) of rotation 
7 
68 
2 
Tetragonal 
Hexagonal 
Trigonal 
1 threefold axis (parallel to z) of rotation 
5 
7 
1 
Rhombohedral 
18 
1 
Hexagonal 
Hexagonal 
1 sixfold axis (parallel to z) of rotation 
7 
27 
Cubic

4 threefold axes of rotation 
5 
36 
3 
Cubic 
Total: 6 
7 

32 
230 
14 
7 
Table 4549c. Symmetry groups for the crystal systems.
System 
Essential symmetry 
Lattice symmetry 
Diffraction (Laue) symmetry 
Point Groups (Hermann–Mauguin notation) 
X 

X⊥m (X/m) 
Xm (Xm) 
( ) 
X⊥2 
X⊥mm 

None 


1 
1 
 
 
 
 
 


2/m 
2/m 
2 
m 
2/m 
 
 
 
 

222 or 2mm

mmm

mmm 
 
 
 
mm2 
 
222 
mmm 


4/mmm

4/m

4 
4 
4/m 
 
 
 
 
4/mmm 
 
 
 
4mm 
42m 
422 
4/mmm 



3 
3 
3 
 
 
 
 
 
3m1 
 
 
 
3m 
3m1 
321 
 
31m 
 
 
 
31m 
312 
 


6/mmm

6/m 
6 
6 
6/m 
 
 
 
 
6/mmm 
 
 
 
6mm 
62m 
622 
6/mmm 

23 
m3m

m3 
23 
m3 
 
 
 
 
 
m3m 
 
 
 
 
43m 
432 
m3m 
*
X: Major Xfold rotation axis alone.
: Inversion axis alone, namely, the major Xfold inversion axes.
X/m: Rotation axis with a symmetry plane normal to it.
Xm: Rotation axis with a symmetry plane that is not normal to it (usually a vertical symmetry plane).
: Inversion axis with a symmetry plane not normal to it.
X⊥2: Rotation axis with a diad normal to it (The twofold rotation axis is perpendicular to the major axis).
X/mm: Rotation axis with a symmetry plane normal to it and another not so. 
Of the 32 point groups, 11 crystal classes are centrosymmetric and thus possess no polar properties. Of the remaining 21 noncentrosymmetric classes, 20 classes show electrical polarity when a stress is applied.
Figure 4549b shows the schematic illustration of the different classes of crystal systems and their properties. In ferroelectric substances, the number of distinct spontaneous polarization directions depends on its point group symmetry.


32 point groups 













Nonpyroelectric 




Nonferroelectric 
Figure 4549b. Schematic illustration of the different classes of crystal systems and their properties.
