K3 SURFACES WITH MAXIMAL FINITE AUTOMORPHISM GROUPS CONTAINING M-20

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is 960 and that if such a group has order 960, then it is isomorphic to the Mathieu group M-20. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is 3840 and this group contains M-20 with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface Km(E-i x E-i) this paper we describe two more K3 surfaces admitting a big finite automorphism group of order 1920, both groups contains M-20 as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.