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A close relationship between K3 surfaces and the Mathieu groups has beenestablished in the last century. Furthermore, it has been observed recentlythat the elliptic genus of K3 has a natural interpretation in terms of thedimensions of representations of the largest Mathieu group M24. In this paperwe first give further evidence for this possibility by studying the ellipticgenus of K3 surfaces twisted by some simple symplectic automorphisms. Thesepartition functions with insertions of elements of M24 (the McKay-Thompsonseries) give further information about the relevant representation. We thenpoint out that this new "moonshine" for the largest Mathieu group is connectedto an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum ofK3xT^2-compactified type II string theory. This insight on the symmetry of thetheory sheds new light on the generalised Kac-Moody algebra structure appearingin the spectrum, and leads to predictions for new elliptic genera of K3,perturbative spectrum of the toroidally compactified heterotic string, and theindex for the 1/4-BPS dyons in the d=4, N=4 string theory, twisted by elementsof the group of stringy K3 isometries.

K3 Surfaces, N=4 Dyons, and the Mathieu Group M24