1. Crystals
Crystal consist of the periodic arrangement of building blocks
Each building block, called a basis, is an atom, a molecule, or a group of atoms or molecules
Such a periodic arrangement must have translational symmetry such that if you move a building block by a distance:
then it falls on another identical building block with the same orientation.
If we remove the building blocks and replace them with points, then we have a point lattice or Bravais lattice.

2. Crystal Symmetry
These Bravais lattices have several symmetry operations (these are operations on the lattice which leave it looking identical to the original lattice).
Translational (as we’ve already seen)
Rotation about an axis (1, 2, 3, 4, or 6 fold)
Reflection through a mirror plane
Inversion through a point
Combination of two of the above
Glide (= reflection + translation)
Screw (= rotation + translation)
Row A a 1 1’ 2
Row B m m’
(m-1) X
For rotational symmetry and translational symmetry:
Note: atom 1 and atom m are separated by a distance (m-1)a

3. Point and Space Groups
32 point groups link
Any group constructed by reducing the symmetry of an object characterized by a particular crystal system continues to belong to that system until the symmetry has been reduced so far that all of the remaining symmetry operations of the object are also found in a less symmetrical crystal system; when this happens the symmetry group of the object is assigned to the less symmetrical system. Thus the crystal system of a crystallographic point group is that of the least symmetric of the seven Bravais lattice point groups containing every symmetry operation of the crystallographic group.
Cubic
Tetragonal
Orthorhombic
Monoclinic
Triclinic
Hexagonal
Trigonal
(Rhombohedral)