Dark matter stabilized by a non-Abelian group: Lessons from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">Σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>36</mml:mn><mml:mo stretchy="f

When building dark matter (DM) models, one often imposes conserved discrete symmetries to stabilize DM candidates. The simplest choice is <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> but models with larger stabilizing groups have also been explored. Can a conserved non-Abelian group lead to a viable DM model? Here, we address this question within the three-Higgs-doublet model based on the group <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="normal">Σ</d:mi><d:mo stretchy="false">(</d:mo><d:mn>36</d:mn><d:mo stretchy="false">)</d:mo></d:math>, in which DM stabilization by a non-Abelian group is not only possible but inevitable. We show that the tight connections between the Higgs, fermion, and DM sectors repeatedly drive the model into conflict with the LHC results and DM observations, with the most recent LZ results playing a decisive role. We believe that the lessons learned from this study help chart the limits of what can be achieved within multi-Higgs-doublet DM models with large symmetry groups.