# ApCoCoA-1:Coxeter groups

From ApCoCoAWiki

#### Description

The full icosahedral group is a Coxeter group. Their relations results of a Matrix, the Coxetermatrix. The Matrix with i lines and j columns gives the following relations:

<r_1,...,r_n|(r_ir_j)^m_ij

-the relation mii means: (r_ir_i)^1=1 for all i

-and other generators r_i, r_j commute.

H3 has the following presentation:

H3 = <x,y,z | x^2 = y^2 = z^2 = (xy)^2 = (xz)^2 =(yz)^2 = 1 >

#### Reference

not found yet

#### Computation

#### Example in Symbolic Data Format

<FREEALGEBRA createdAt="2014-07-30" createdBy="strohmeier"> <vars>x,y,z</vars> <basis> <ncpoly>x*x-1</ncpoly> <ncpoly>y*y-1</ncpoly> <ncpoly>z*z-1</ncpoly> <ncpoly>(x*y)^5-1</ncpoly> <ncpoly>(y*x)^5-1</ncpoly> <ncpoly>(x*z)^2-1</ncpoly> <ncpoly>(z*x)^2-1</ncpoly> <ncpoly>(y*z)^3-1</ncpoly> <ncpoly>(z*y)^3-1</ncpoly> </basis> <Comment>Coxeter_Group_H3</Comment> </FREEALGEBRA>