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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. Depending on their geometry, crystals are commonly classified into seven systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. Figure 4548a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. Table 4548a also lists the relation between threedimensional crystal families, crystal systems, and lattice systems. Table 4548b also shows the seven crystal systems and the restrictions on their cell dimensions (cell edges and cell angles).
Figure 4548a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.
Table 4548a. The relation between threedimensional crystal families, crystal systems, and lattice systems.
Crystal
family 
Crystal
system 
Required
symmetries
of point group 
Point
group 
Space
group 
Bravais
lattices 
Lattice
system 
Triclinic 
None 
2 
2 
1 
Triclinic 
Monoclinic

1 twofold axis 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 of rotation 
7 
68 
2 
Tetragonal 
Hexagonal 
Trigonal 
1 threefold axis of rotation 
5 
7 
1 
Rhombohedral 
18 
1 
Hexagonal 
Hexagonal 
1 sixfold axis of rotation 
7 
27 
Cubic

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

32 
230 
14 
7 
As indicated in Table 4548a, the 14 basic Bravais lattice types give rise to a total of 230 possible crystal types when they are combined with other symmetry operators such as mirrors planes, glide planes, and screw axes.
Table 4548b. The seven crystal systems (families) and the restrictions on their cell dimensions.
Crystal family 
Cell edges 
Cell angles 
Triclinic 
None 
None 
Monoclinic 
None 
α = γ = 90° 
Orthorhombic 
None 
α = β = γ = 90° 
Tetragonal 
a = b 
α = β = γ = 90° 
Trigonal, hexagonal 
a = b 
α = β = 90°, γ = 120° 
Cubic 
a = b = c 
α = β = γ = 90° 
Figure 4548b shows the schematic illustration of the different classes of crystal systems and their properties.
















Nonpyroelectric 




Nonferroelectric 
Figure 4548b. Schematic illustration of the different classes of crystal systems and their properties.
