Remark 2

Let G be a finite group generated by reflections and essential. Then G is crystallographic if and only if every element of its Coxeter matrix is one of the integers $1, 2, 3, 4, 6$. Indeed, this condition is necessary by Remark 3) of § 1, no. 5. The fact that it is sufficient will follow from the classification of finite Coxeter groups given in § 4 (for a direct proof, see Chap. V, § 4, Exerc. 6).