Cohomology of the Double Cover of the Mathieu Group M12

In this paper we calculate the mod two cohomology of the double cover of the Mathieu group M12. The starting point is the calculation by Adem, Maginnis and Milgram of the mod two cohomology of M12. We use a hypercohomology spectral sequence to determine a differential in the Lyndon-Hochschild-Serre spectral sequence of the central extension, and this gets us as far as the E∞ page. Ungrading requires restriction to the Sylow 2-subgroup, and here some computer calculations come to our rescue.