\title{Periodic resolutions for certain finite groups}
\author{Dave Benson}
\address{Department of Mathematics, University of Aberdeen, 
Aberdeen AB24 3UE, Scotland.}
\begin{abstract}
Let $p$ and $q$ be distinct odd primes, and let 
\[ G = \langle x,y\mid x^py=y^{-1}x^p,\ xy=y^{2q-1}x^{-1}\rangle 
   \cong \Z/pq\rtimes Q_8. \] 
Using this deficiency zero presentation discovered by Bernard Neumann, 
we investigate the beginning of a projective 
resolution of $\Z$ as a $\Z G$-module in the case $p\not\equiv q \pmod 4$. 
This gives enough information
to compute $H^*(G,M)$ for any $\Z G$-module $M$.
\end{abstract}