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We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group  4 . Our main tool is a new presentation for the algebra C( 4 ), corresponding to an isomorphism of type  4  ≃  SO  –1 (3). The quantum subgroups of  4  are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

Quantum groups acting on 4 points