material taken from (MIF)
By classical results due to Nikulin, Mukai, Xiao and Kondo in the 1980’s and 90’s, the finite symplectic automorphism groups of K3 surfaces are always subgroups of the Mathieu group $M_{24}$. This is a sporadic finite simple group of order 244823040. However, also by results due to Mukai, each such automorphism group has at most 394 elements and thus is by orders of magnitude smaller than $M_{24}$. On the other hand, according to a recent observation by Eguchi, Ooguri and Tachikawa, the elliptic genus of K3 surfaces seems to contain a mysterious footprint of an action of the entire group $M_{24}$: If one decomposes the elliptic genus into irreducible characters of the N=4 superconformal algebra, which is natural in view of superconformal field theories (SCFTs) associated to K3, then the coefficients of the so-called non-BPS characters coincide with the dimensions of representations of $M_{24}$.
Mathieu moonshine corresponds to one of the 23 Neimeier lattices in the generalization to all such lattices known as umbral moonshine.
Matthias Gaberdiel, Roberto Volpato, Mathieu Moonshine and Orbifold K3s (arXiv:1206.5143)
Anne Taormina, Katrin Wendland, The overarching finite symmetry group of Kummer surfaces in the Mathieu group $M_24$ (arXiv:1107.3834)
Terry Gannon, Much ado about Mathieu, (arXiv:1211.5531)
Robert Volpato, Mathieu moonshine and elliptic genus of K3 (2010) (pdf)
Mathematisches Institute Freiburg, Mathieu moonshine (web)