Minimal semi-annihilating ${\mathbb{Z}}_N$ scalar dark matter

We study the dark matter from an inert doublet and a complex scalar singlet stabilized by ${\mathbb{Z}}_N$ symmetries. This field content is the minimal one that allows dimensionless semi-annihilation couplings for $N>2$. We consider explicitly the ${\mathbb{Z}}_3$ and ${\mathbb{Z}}_4$ cases and take into account constraints from perturbativity, unitarity, vacuum stability, necessity for the electroweak ${\mathbb{Z}}_N$ preserving vacuum to be the global minimum, electroweak precision tests, upper limits from direct detection and properties of the Higgs boson. Co-annihilation and semi-annihilation of dark sector particles as well as dark matter conversion significantly modify the cosmic abundance and direct detection phenomenology.