gt.geometric topology - Free actions of finite groups on products of even-dimensional spheres

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 Q₈ or the dihedral group Dih₄ 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.

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