  We study a 3d lattice gauge theory with gauge group
$\mathrm{U}(1)^{N-1}\rtimes \mathrm{S}_N$, which is obtained by gauging the
$\mathrm{S}_N$ global symmetry of a pure $\mathrm{U}(1)^{N-1}$ gauge theory,
and we call it the semi-Abelian gauge theory. We compute mass gaps and string
tensions for both theories using the monopole-gas description. We find that the
effective potential receives equal contributions at leading order from
monopoles associated with the entire $\mathrm{SU}(N)$ root system. Even though
the center symmetry of the semi-Abelian gauge theory is given by
$\mathbb{Z}_N$, we observe that the string tensions do not obey the $N$-ality
rule and carry more detailed information on the representations of the gauge
group. We find that this refinement is due to the presence of non-invertible
topological lines as a remnant of $\mathrm{U}(1)^{N-1}$ one-form symmetry in
the original Abelian lattice theory. Upon adding charged particles
corresponding to $W$-bosons, such non-invertible symmetries are explicitly
broken so that the $N$-ality rule should emerge in the deep infrared regime.
