Suppose a finite 2-group *G* acts freely on *X* = $\prod_{i=1}^k$ *S*$^{2n_i}$, a product of *k* even-dimensional spheres, *k* > 2. Is it possible for *G* to be non-abelian? What if we additionally assume that spheres in the product are equidimensional?

Some comments: The equality 2^{k} = $\chi(X)$ = |G|$\chi(X/G)$ coming from the covering *X* $\to$ *X/G* ensures that a finite group acting freely on *X* is a 2-group and also answers the question for *k* = 1, 2. (Actually, for *k* = 1 this gives a proof of a classical theorem: the only group which can act freely on an even-dimensional sphere is the cyclic group of order 2. Does anyone know who is this result originally due to? Sorry for a question inside the question; perhaps someone can comment on this one.) Hence if one would like to construct a sort of "minimal" example, it should involve an action of either the quaternion group Q_{8} or the dihedral group Dih_{4} for *k* = 3. I thought about this a little bit, but I feel like I'm not comfortable enough with non-abelian groups.

I've stumbled across some papers where authors characterize arbitrary finite groups acting freely on *X* in terms of existence of particular representations, but explicit examples are given only in the abelian case.