Coxeter group
The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a linear representation of a Coxeter group. For "finite" reflection groups, this yields an exact correspondence: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter groups, however, a Coxeter group may not admit a representation as a reflection group.