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In this note we interpret a recent result of Gaberdiel, Hohenegger andVolpato in terms of derived equivalences of K3 surfaces. We prove that there isa natural bijection between subgroups of the Conway group Co_1 with invariantlattice of rank at least four and groups of symplectic derived equivalences ofD(Coh(X)) of projective K3 surfaces fixing a stability condition.

On derived categories of K3 surfaces, symplectic automorphisms and the Conway group