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We study the dark matter from an inert doublet and a complex scalar singletstabilized by $\mathbb{Z}_N$ symmetries. This field content is the minimal onethat allows dimensionless semi-annihilation couplings for $N > 2$. We considerexplicitly the $\mathbb{Z}_3$ and $\mathbb{Z}_4$ cases and take into accountconstraints from perturbativity, unitarity, vacuum stability, necessity for theelectroweak $\mathbb{Z}_N$ preserving vacuum to be the global minimum,electroweak precision tests, upper limits from direct detection and propertiesof the Higgs boson. Co-annihilation and semi-annihilation of dark sectorparticles as well as dark matter conversion significantly modify the cosmicabundance and direct detection phenomenology.

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