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Symmetry describes how a pattern repeats within a crystal. 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.
Symmetry operators are the motions that allow a pattern to be transformed from an initial position to a final position and the initial and final patterns are indistinguishable. The symmetry operators are:
i) Translation.
ii) Reflection.
iii) Rotation.
iv) Inversion (center of symmetry).
v) Rotoinversion (Rotation followed by inversion through the origin).
vi) Rotoreflection.
vii) Glide (translation plus reflection).
viii) Screw (rotation plus translation).
Figure 3556 schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. Table 3556a also lists the symmetry operations and symmetry elements. Table 3556b shows the relation between threedimensional crystal families, crystal systems, and lattice systems.
Figure 3556. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.
Table 3556a. Symmetry operations and symmetry elements

Symmetry element 
Symmetry operation 
E 

Identity ^{[1]} 
C_{n} 
nfold symmetry axis 
Rotation by 2π/n 
σ 
Mirror plane 
Reflection 
i 
Center of inversion 
Inversion 
S_{n} 
nfold axis of improper rotation ^{[2]} 
Rotation by 2π/n followed by reflection perpendicular to rotation axis 
[1] The symmetry element can be thought of as the whole of space.
[2] The equivalences S_{1} = σ and S_{2} = i.
Table 3556b. 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 
With stereographic projections, we can easily visualize crystallographic features:
i) Crystal symmetry.
ii) Slip planes and directions.
iii) Crystal planes and orientation relationships.
iv) Grain orientations.
