We study Coxeter groups from which there is a natural map onto a symmetric group. Such
groups have natural quotient groups related to presentations of the symmetric group on an
arbitrary set T of transpositions. These quotients, denoted here by CY (T), are a special case
of the generalized Coxeter groups defined in [5], and also arise in the computation of certain
invariants of surfaces. We use a surprising action of Sn on the kernel of the surjection CY
(T)→ Sn to show that this kernel embeds in the direct product of n copies of the free group π …