  Fixed points in three dimensions described by conformal field theories with
$MN_{m,n}= O(m)^n\rtimes S_n$ global symmetry have extensive applications in
critical phenomena. Associated experimental data for $m=n=2$ suggest the
existence of two non-trivial fixed points, while the $\varepsilon$ expansion
predicts only one, resulting in a puzzling state of affairs. A recent numerical
conformal bootstrap study has found two kinks for small values of the
parameters $m$ and $n$, with critical exponents in good agreement with
experimental determinations in the $m=n=2$ case. In this paper we investigate
the fate of the corresponding fixed points as we vary the parameters $m$ and
$n$. We find that one family of kinks approaches a perturbative limit as $m$
increases, and using large spin perturbation theory we construct a large $m$
expansion that fits well with the numerical data. This new expansion, akin to
the large $N$ expansion of critical $O(N)$ models, is compatible with the fixed
point found in the $\varepsilon$ expansion. For the other family of kinks, we
find that it persists only for $n=2$, where for large $m$ it approaches a
non-perturbative limit with $\Delta_\phi\approx 0.75$. We investigate the
spectrum in the case $MN_{100,2}$ and find consistency with expectations from
the lightcone bootstrap.
