  We undertake a systematic study of the $4$-dimensional $SU(N)$ $2$-index
chiral gauge theories and investigate their faithful global symmetries and
dynamics. These are a finite set of theories with fermions in the $2$-index
symmetric and anti-symmetric representations, with no fundamentals, and they do
not admit a large-$N$ limit. We employ a combination of perturbative and
nonperturbative methods, enabling us to constrain their infrared (IR) phases.
Specifically, we leverage the 't Hooft anomalies associated with continuous and
discrete groups to eliminate a few scenarios. In some cases, the anomalies rule
out the possibility of fermion composites. In other cases, the interplay
between the continuous and discrete anomalies leads to multiple higher-order
condensates, which inevitably form to match the anomalies. Further, we pinpoint
the most probable symmetry-breaking patterns by searching for condensates that
match the full set of anomalies resulting in the smallest number of IR degrees
of freedom. Higher-loop $\beta$-function analysis suggests that a few theories
may flow to a conformal fixed point.
