**Discrete Subgroups**

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.

Other finite subgroups include:

- Permutation matrices (the Coxeter group A
_{n}) - Signed permutation matrices (the Coxeter group B
_{n}); also equals the intersection of the orthogonal group with the integer matrices.

Read more about this topic: Orthogonal Group, Related Groups

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