In Euclidean geometry, discrete symmetry groups come in two types: finite **point groups**, which include only rotations and reflections, and infinite **lattice groups**, which also include translations and glide reflections. There are also continuous symmetry groups, which are Lie groups.

**Two dimensions**

The two simplest point groups in 2-D space are the trivial group *C*_{1}, where no symmetry operations leave the object unchanged, and the group containing only the identity and reflection about a particular line, *D*_{1}. The other point groups form two infinite series, called *C _{n}* and

Examples:

*** *** *** * ** * * * *** * * * *** * CGroups including translation in a single direction are called frieze groups. There are seventeen 2-D lattice groups including translation in multiple directions, called wallpaper groups._{1}D_{1}C_{2}D_{4}

**Three dimensions**

The situation in 3-D is more complicated, since it is possible to have multiple rotation axes in a point group. First, of course, there is the trivial group, and then there are three groups of order 2, called *C _{s}*,

The last of these is the first of the uniaxial groups *C _{n}*, which are generated by a single rotation of angle 2π/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group

This new group is called *D _{nh}*, and its subgroup of rotations is

There is one more group in this family to mention, called *S _{n}*. This group is generated by an improper rotation of angle 2π/n - that is, a rotation followed by a reflection about a plane perpendicular to its axis. For n even, the rotation and reflection are generated, so this becomes the same as

The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using C_{n} to denote an axis of rotation through 2π/n and S_{n} to denote an axis of improper rotation through the same, the groups are:

**T**. There are four C_{3} axes, directed through the corners of a cube, and three C_{2} axes, directed through the centers of its faces. There are no other symmetry operations, giving the group an order of 12.

**T _{d}**. This group has the same rotation axes as

**T _{h}**. This group has the same rotation axes as

**O**. This group is similar to *T*, but the C_{2} axes are now C_{4} axes, and a new set of 12 C_{2} axes appear, directed towards the edges of the original cube. Another group of order 24...

**O _{h}**. This group has the same rotation axes as

**I**, **I _{h}** are the groups of symmetries and the group of rotations of the icosahedron (also of the dodecahedron and icosidodecahedron). The group of rotations is a normal subgroup of index 2 in the group of symmetries, with