  We initiate a numerical conformal bootstrap study of CFTs with $S_n \ltimes
(S_Q)^n$ global symmetry. These include CFTs that can be obtained as coupled
replicas of two-dimensional critical Potts models. Particular attention is paid
to the special case $S_3 \ltimes (S_3)^3$, which governs the critical behaviour
of three coupled critical 3-state Potts models, a multi-scalar realisation of a
(potentially) non-integrable CFT in two dimensions. The model has been studied
in earlier works using perturbation theory, transfer matrices, and Monte Carlo
simulations. This work represents an independent non-perturbative analysis. Our
results are in agreement with previous determinations: we obtain an allowed
peninsula within parameter space for the scaling dimensions of the three
lowest-lying operators in the theory, which contains the earlier predictions
for these scaling dimensions. Additionally, we derive numerous bounds on
admissible scaling dimensions in the theory, which are compatible with earlier
results. Our work sets the necessary groundwork for a future precision study of
these theories in the conformal bootstrap.
