Editing Coxeter group (section)

===Symmetry groups of regular polytopes===
The symmetry group of every regular polytope is a finite Coxeter group. Note that [[dual polytope]]s have the same symmetry group.

There are three series of regular polytopes in all dimensions. The symmetry group of a regular ''n''-simplex is the symmetric group ''S''<sub>''n''+1</sub>, also known as the Coxeter group of type ''A<sub>n</sub>''. The symmetry group of the ''n''-[[cube]] and its dual, the ''n''-cross-polytope, is ''B<sub>n</sub>'', and is known as the [[hyperoctahedral group]].

The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, the [[dihedral group]]s, which are the symmetry groups of [[regular polygon]]s, form the series ''I''<sub>2</sub>(''p''), for ''p'' ≥ 3. In three dimensions, the symmetry group of the regular [[dodecahedron]] and its dual, the regular [[icosahedron]], is ''H''<sub>3</sub>, known as the [[full icosahedral group]]. In four dimensions, there are three exceptional regular polytopes, the [[24-cell]], the [[120-cell]], and the [[600-cell]]. The first has symmetry group ''F''<sub>4</sub>, while the other two are dual and have symmetry group ''H''<sub>4</sub>.

The Coxeter groups of type ''D''<sub>''n''</sub>, ''E''<sub>6</sub>, ''E''<sub>7</sub>, and ''E''<sub>8</sub> are the symmetry groups of certain [[semiregular polytope]]s.