2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/215927We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group $\mathcal Q_4$. Our main tool is a new presentation for the algebra $\rm C(\mathcal Q_4)$, corresponding to an isomorphism of type $\mathcal Q_4\simeq SO_{-1}(3)$. The quantum subgroups of $\mathcal Q_4$ are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.34 pages. Comments added in last sectionQuantum AlgebraGroup Theory20G42; 16W30text