  We discuss gravitational waves in an electroweakly interacting vector dark
matter model. In the model, the electroweak gauge symmetry is extended to
SU(2)$_0 \times$ SU(2)$_1 \times$SU(2)$_2 \times$ U(1)$_Y$ and spontaneously
broken into SU(2)$_L \times$ U(1)$_Y$ at TeV scale. The model has an exchange
symmetry between SU(2)$_0$ and SU(2)$_2$. This symmetry stabilizes some massive
vector bosons associated with the spontaneous symmetry breaking described
above, and an electrically neutral one is a dark matter candidate. In the
previous study, it was found that the gauge couplings of SU(2)$_0$ and
SU(2)$_1$ are relatively large to explain the measured value of the dark matter
energy density via the freeze-out mechanism. With the large gauge couplings,
the gauge bosons potentially have a sizable effect on the scalar potential. In
this paper, we focus on the phase transition of SU(2)$_0 \times$ SU(2)$_1
\times$ SU(2)$_2 \to$ SU(2)$_L$. We calculate the effective potential at finite
temperature and find that the phase transition is first-order and strong in a
wide range of the parameter space. The strong first-order phase transition
generates gravitational waves. We calculate the gravitational wave spectrum and
find that it is possible to detect the gravitational waves predicted in the
model by future space-based gravitational wave interferometers. We explore the
regions of the parameter space probed by the gravitational wave detection. We
find that the gravitational wave detection can probe the region where the mass
of $h'$, a CP-even scalar in the model, is a few TeV.
