  A close relationship between K3 surfaces and the Mathieu groups has been
established in the last century. Furthermore, it has been observed recently
that the elliptic genus of K3 has a natural interpretation in terms of the
dimensions of representations of the largest Mathieu group M24. In this paper
we first give further evidence for this possibility by studying the elliptic
genus of K3 surfaces twisted by some simple symplectic automorphisms. These
partition functions with insertions of elements of M24 (the McKay-Thompson
series) give further information about the relevant representation. We then
point out that this new "moonshine" for the largest Mathieu group is connected
to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of
K3xT^2-compactified type II string theory. This insight on the symmetry of the
theory sheds new light on the generalised Kac-Moody algebra structure appearing
in the spectrum, and leads to predictions for new elliptic genera of K3,
perturbative spectrum of the toroidally compactified heterotic string, and the
index for the 1/4-BPS dyons in the d=4, N=4 string theory, twisted by elements
of the group of stringy K3 isometries.
