Alternating subgroups of Coxeter groups and their spinor extensions

Let G be a discrete Coxeter group, G+ its alternating subgroup and the spinor cover of G+. A presentation of the groups G+ and is proved for an arbitrary Coxeter system (G,S); the generators are related to edges of the Coxeter graph. Results of the Coxeter–Todd algorithm–with this presentation–for the chains of alternating groups of types A, B and D are given.