  We carry out a systematic study of $SU(6)$ Yang-Mills theory endowed with
fermions in the adjoint and $3$-index antisymmetric mixed-representation. The
fermion bilinear in the $3$-index antisymmetric representation vanishes
identically, which leads to interesting new phenomena. We first study the
theory on a small circle, i.e., on $\mathbb R^3\times \mathbb S^1_L$, employing
symmetry-twisted boundary conditions and semi-classical techniques. We find
that the ground state is $3$-fold degenerate, which can be explained as a
consequence of a $1$-form/$0$-form mixed 't Hooft anomaly. In addition, the
theory may admit massless bosonic and fermionic degrees of freedom, depending
on the number of flavors, and confines the electric probes in the infrared.
Empowered by 't Hooft anomaly matching conditions along with the $2$-loop
$\beta$-function, we further examine the possible infrared symmetry
realizations on $\mathbb R^4$ for various number of adjoint and $3$-index
antisymmetric fermions. The infrared theory is either a conformal field theory,
which is expected for a large number of flavors, or it is confining with or
without chiral symmetry breaking. In a few cases, we are able to give enough
evidence for adiabatic continuity between the small- and large-circle limits.
