  We investigate the representation of a symmetric group $S_n$ on the homology
of its Quillen complex at a prime $p$. For homology groups in small
codimension, we derive an explicit formula for this representation in terms of
the representations of symmetric groups on homology groups of $p$-uniform
hypergraph matching complexes. We conjecture an explicit formula for the
representation of $S_n$ on the top homology group of the corresponding
hypergraph matching complex when $n \equiv 1 \bmod p$. Our conjecture follows
from work of Bouc when $p=2$, and we prove the conjecture when $p=3$.

Group Theory;Combinatorics