  We propose a model with two Higgs doublets and several $SU(2)$ scalar
singlets with a global non-Abelian flavor symmetry
$\mathcal{Q}_6\times\mathcal{Z}_2$. This discrete group accounts for the
observed pattern of fermion masses and mixing angles after spontaneous symmetry
breaking. In this scenario only the third generation of fermions get their
masses as in the Standard Model (SM). The masses of the remaining fermions are
generated through a seesaw-like mechanism. To that end, the matter content of
the model is enlarged by introducing electrically charged vector-like fermions
(VLFs), right handed Majorana neutrinos and several SM scalar singlets. Here we
study the processes involving VLFs that are within the reach of the Large
Hadron Collider (LHC). We perform collider studies for vector-like leptons
(VLLs) and vector-like quarks (VLQs), focusing on double production channels
for both cases, while for VLLs single production topologies are also included.
Utilizing genetic algorithms for neural network optimization, we determine the
statistical significance for a hypothetical discovery at future LHC runs. In
particular, we show that we can not safely exclude VLLs for masses greater than
$200~\mathrm{GeV}$. For VLQ's in our model, we show that we can probe their
masses up to 3.8 TeV, if we take only into account the high-luminosity phase of
the LHC. Considering Run-III luminosities, we can also exclude VLQs for masses
up to $3.4~\mathrm{TeV}$. We also show how the model with predicted VLL masses
accommodates the muon anomalous magnetic moment.
