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Systematic analysis of 3HDM symmetries

Symmetries play a crucial role in shaping the structure and predictions of multi-Higgs-doublet models. In three-Higgs-doublet models considerable effort has been put into classifying possible symmetry groups and the conditions for their realisation, yet the completeness of existing classifications remains an open question. In this work, we revisit the problem of identifying realisable symmetries by re-examining conventional Higgs family and general CP transformations from an alternative perspective. Our analysis identifies certain limitations in previous approaches and introduces a clearer, more systematic framework for model builders. We expand our classification by investigating more generalised symmetry structures -- the recently identified GOOFy transformations, which act non-trivially on the Higgs doublets and their conjugates. Our analysis consolidates known results, uncovers previously overlooked structures, and expands the set of symmetries in three-Higgs-doublet models, offering both a clearer theoretical foundation and a practical reference for symmetry-based model building.