Dark matter stabilized by a non-Abelian group

Viable dark matter (DM) models can be built with the addition of conserved discrete symmetries that stabilize DM candidates. The simpler choice of symmetry is the [Formula: see text] but models with larger stabilizing groups can also be considered. A question then is if a conserved non-Abelian group can lead to a viable DM model? We attempt to answer the question within the three-Higgs-doublet model based on the group [Formula: see text], in which the DM stabilization by a non-Abelian group is inevitable. We show that the tension between the Higgs, fermion, and DM sectors repeatedly drives the model into conflict with the LHC results and DM observations, with a highlight on the most recent LZ results. We believe that this study helps chart the limits of what can be achieved within multi-Higgs-doublet DM models with large symmetry groups.