  Let $G$ be a discrete Coxeter group, $G^+$ its alternating subgroup and
$\tilde{G}^+$ the spinor cover of $G^+$. A presentation of the groups $G^+$ and
$\tilde{G}^+$ 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.
