  We investigate a perturbatively renormalizable $S_{q}$ invariant model with
$N=q-1$ scalar field components below the upper critical dimension
$d_c=\frac{10}{3}$. Our results hint at the existence of multicritical
generalizations of the critical models of spanning random clusters and
percolations in three dimensions. We also discuss the role of our multicritical
model in a conjecture that involves the separation of first and second order
phases in the $(d,q)$ diagram of the Potts model.
