Finite imprimitive rank 3 affine groups – I

Abstract This is one of a series of papers which aims to classify imprimitive affine groups of rank 3. In this paper, a complete classification is given of such groups of characteristic 𝑝 whose point stabilizers are not 𝑝-local, which shows that such groups are very rare; specifically, the only examples are the two non-isomorphic groups of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mn>4</m:mn> </m:msup> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> 2^{4}{:}\mathrm{GL}_{3}(2) with a unique minimal normal subgroup.