  We study $4$-dimensional $SU(N)\times U(1)$ gauge theories with a single
massless Dirac fermion in the $2$-index symmetric/antisymmetric representations
and show that they are endowed with a noninvertible $0$-form $\widetilde
{\mathbb Z}_{2(N\pm 2)}^{\chi}$ chiral symmetry along with a $1$-form $\mathbb
Z_N^{(1)}$ center symmetry. By using the Hamiltonian formalism and putting the
theory on a spatial three-torus $\mathbb T^3$, we construct the non-unitary
gauge invariant operator corresponding to $\widetilde {\mathbb Z}_{2(N\pm
2)}^{\chi}$ and find that it acts nontrivially in sectors of the Hilbert space
characterized by selected magnetic fluxes. When we subject $\mathbb T^3$ to
$\mathbb Z_N^{(1)}$ twists, for $N$ even, in selected magnetic flux sectors,
the algebra of $\widetilde {\mathbb Z}_{2(N\pm 2)}^{\chi}$ and $\mathbb
Z_N^{(1)}$ fails to commute by a $\mathbb Z_2$ phase. We interpret this
noncommutativity as a mixed anomaly between the noninvertible and the $1$-form
symmetries. The anomaly implies that all states in the torus Hilbert space with
the selected magnetic fluxes exhibit a two-fold degeneracy for arbitrary
$\mathbb T^3$ size. The degenerate states are labeled by discrete electric
fluxes and are characterized by nonzero expectation values of condensates. In
an Appendix, we also discuss how to construct the corresponding noninvertible
defect via the ``half-space gauging'' of a discrete one-form magnetic symmetry.
