Varieties isogenous to a higher product with prescribed numerical invariants

Abstract Using structural properties of groups of small order, we establish the non-existence of varieties isogenous to a higher product of dimension n greater than 3 with fixed topological Euler number $$(-2)^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> and trivial first Betti number.