  We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating
the Mathieu group M_24 and the complex elliptic genus of a K3 surface with the
symmetries of geometric structures on K3 surfaces.
  Two main results are that the complex elliptic genus of a K3 surface is a
virtual module for the Mathieu group M_24 and also for a certain vertex
operator superalgebra V^G where G is the holonomy group.
