\title{Cohomology of the double cover of the Mathieu group $M_{12}$.}
\author{D. J. Benson}
\address{Dept. of Mathematics, University of Georgia, Athens GA 30602, USA}
\thanks{Both authors are partly supported by grants from the NSF}
\author{J. F. Carlson}
\address{Dept. of Mathematics, University of Georgia, Athens GA 30602, USA}
\begin{abstract}
In this paper we calculate the mod two cohomology of the double cover
of the Mathieu group $M_{12}$. The starting point is the calculation
by Adem, Maginnis and Milgram of the mod two cohomology of $M_{12}$.
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 $E_\infty$ page. Ungrading
requires restriction to the Sylow $2$-subgroup, and here some computer
calculations come to our rescue.
\end{abstract}