  There is a `Mathieu moonshine' relating the elliptic genus of K3 to the
sporadic group M_{24}. Here, we give evidence that this moonshine extends to
part of the web of dualities connecting heterotic strings compactified on K3
\times T^2 to type IIA strings compactified on Calabi-Yau threefolds. We
demonstrate that dimensions of M_{24} representations govern the new
supersymmetric index of the heterotic compactifications, and appear in the
Gromov--Witten invariants of the dual Calabi-Yau threefolds, which are elliptic
fibrations over the Hirzebruch surfaces F_n.
