Hypergraph matching complexes and Quillen complexes of symmetric groups

We investigate Quillen complexes of symmetric groups using hypergraph matching complexes and p-cycle complexes. We determine the homotopy type of the Quillen complex of S3p at the prime p. We show that there is torsion in the homology of the Quillen complex of S13 at the prime 3, thereby providing the first example of a Quillen complex of a finite group whose homology is not free.