  Theories in which the dark matter (DM) candidate is a fermion transforming
chirally under a gauge symmetry are attractive, as the gauge symmetry would
protect the DM mass. In such theories, the universe would have undergone a
phase transition at early times that generated the DM mass upon spontaneous
breaking of the gauge symmetry. In this paper, we explore the gravitational
wave signals of a simple such theory based on an $\mathrm{SU}(2)_\mathrm{D}$
dark sector with a dark isospin-$3/2$ fermion serving as the DM candidate. This
is arguably the simplest chiral theory possible. The scalar sector consists of
a dark isospin-$3$ multiplet which breaks the $\mathrm{SU}(2)_\mathrm{D}$ gauge
symmetry and also generates the DM mass. We construct the full thermal
potential of the model and identify regions of parameter space which lead to
detectable gravitational wave signals, arising from a strong first-order
$\mathrm{SU}(2)_\mathrm{D}$ phase transition, in various planned space-based
interferometers, while also being consistent with dark matter relic abundance.
Bulk of the parameter space exhibiting detectable gravitational wave signals in
the model also has large WIMP-nucleon scattering cross sections, $\sigma_{\rm
SI}$, which could be probed in upcoming direct detection experiments.
