  We show that certain quotients of the compactified moduli space of $n-$
pointed genus $g$ curves, $\overline{\mathcal{M}}^G:=
\overline{\mathcal{M}}_{g,n} / G$, are of general type, for a fairly broad
class of subgroups $G$ of the symmetric group $S_n$ which act by permuting the
$n$ marked points. The values of $(g,n)$ which we specify in our theorems are
near optimal in the sense that, at least in he cases that G is the full
symmetric group $S_n$ or a product $S_{n_1}\times \ldots \times S_{n_m}$, there
is a relatively narrow transitional zone in which $\overline{\mathcal{M}}^G$
changes its behaviour from being of general type to its opposite, e.g. being
uniruled or even unirational. As an application we consider the universal
difference variety $\overline{\mathcal{M}}_{g,2n} /S_n \times S_n$.

Algebraic Geometry