Both dihedral groups and symmetric groups belong to a larger class of groups, called finite Coxeter groups, which are characterized by the fact that they have a particularly nice and homogeneous presentation. These groups frequently occur as controlling structures of other families of algebraic objects and are therefore studied again and again, from different angles. In this talk I will demonstrate how such groups can be represented by suitable data structures in a computer algebra system like GAP. Furthermore, I will discuss how experiments can be carried out in this setup, both for the purpose of illustrating known properties of finite Coxeter groups with concrete examples, and for the purpose of discovering and formulating new results.