  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 C_Y(T), are a special type of the generalized Coxeter groups
defined in \cite{CST}, and also arise in the computation of certain invariants
of surfaces.
  We use a surprising action of $S_n$ on the kernel of the surjection $C_Y(T)
\ra S_n$ to show that this kernel embeds in the direct product of $n$ copies of
the free group $\pi_1(T)$ (with the exception of $T$ being the full set of
transpositions in $S_4$). As a result, we show that the groups $C_Y(T)$ are
either virtually Abelian or contain a non-Abelian free subgroup.
