  The aim of this paper is to study the geometry of the stack of
$S_{3}$-covers. We show that it has two irreducible components
$\mathcal{Z}_{S_{3}}$ and $\mathcal{Z}_{2}$ meeting in a "degenerate" point
$\{0\}$, $\mathcal{Z}_{2}-\{0\}\simeq \rm B\rm{GL}_{2}$, while
$(\mathcal{Z}_{S_{3}}-\{0\})$, which contains $\rm B S_{3}$ as open substack,
is a smooth and universally closed algebraic stack. More precisely we show that
$\mathcal{Z}_{S_{3}}-\{0\}\simeq[X/\rm{GL}_{2}]$, where $X$ is an explicit
smooth non degenerate projective surface inside $\mathbb{P}^{7}$ intersection
of five quadrics. All these results are based on the description of certain
families of $S_{3}$-covers in terms of "building data".
