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We consider the Cartan subalgebra of any very extended algebra G +++ where G is a simple Lie algebra and let the parameters be space-time fields. These are identified with diagonal metrics and dilatons. Using the properties of the algebra, we find that for all very extensions G +++ of simple Lie algebras there are theories of gravity and matter, which admit classical solutions carrying representations of the Weyl group of G +++ . We also identify the T and S-dualities of superstrings and of the bosonic string with Weyl reflections and outer automorphisms of well-chosen very extended algebras and we exhibit specific features of the very extensions. We take these results as indication that very extended algebras underlie symmetries of any consistent theory of gravity and matter, and might encode basic information for the construction of such theory.

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In view of a potential interpretation of the role of the Mathieu group M 24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different K3 surfaces to larger 'overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type A 24 1 , which is simultaneously compatible with the finite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice serves to express all these symplectic automorphisms as elements of the Mathieu group M 24 , generating the 'overarching finite symmetry group' (Z 2 ) 4 ⋊ A 7 of Kummer surfaces. This group has order 40320, thus surpassing the size of the largest finite symplectic automorphism group of a K3 surface by orders of magnitude. For every Kummer surface this group contains the group of symplectic automorphisms leaving the Kähler class invariant which is induced from the underlying torus. Our results are in line with the existence proofs of Mukai and Kondo, that finite groups of symplectic automorphisms of K3 are subgroups of one of eleven subgroups of M 23 , and we extend their techniques of lattice embeddings for all Kummer surfaces with Kähler class induced from the underlying torus.

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