  The recent emergence of the modern conformal bootstrap method for the study
of conformal field theories (CFTs) has enabled the revisiting of old problems
in classical critical phenomena described by three-dimensional CFTs. The study
of such CFTs with $O(m)^n \rtimes S_n$ global symmetry, also known as MN
models, is pursued in this work. Systems of mixed correlators involving scalar
operators in two different representations of the global symmetry group are
considered. Isolated allowed regions are found in parameter space for various
values of $m$ and $n$. These "islands" can be separated into two qualitative
groups: those close to the unitarity bound and those further away. As a
by-product of our analysis generic tensor structures required to bootstrap any
$G^n \rtimes S_n$ theory with $G$ arbitrary are worked out.
