Cohomology Classes of the Qd(p) Groups

The Qd(p) groups arise in the study of a well-known conjecture by Benson and Carlson, stating that a finite group of rank n acts freely on a finite complex homotopy equivalent to a product of n spheres. These rank-2 groups are noteworthy in that they lack a particular cohomology class, an effective Euler class, which is used to construct free actions on products of two spheres. The present work describes the modular and integral cohomology of Qd(p) in terms of the cohomology of its Sylow-p subgroup, the extra-special p-group of order p3 and exponent p, for p>3. © The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2023.