Finite imprimitive rank $3$ affine groups -- I

TL;DR

This paper classifies certain imprimitive affine groups of rank 3 with specific characteristics, revealing that such groups are extremely rare, with only two known examples fitting the criteria.

Contribution

It provides a complete classification of rank 3 imprimitive affine groups with non-p-local point stabilizers in characteristic p, identifying only two such groups.

Findings

Only two non-isomorphic groups of the form 2^4{:}GL_3(2) exist with the specified properties.

Such groups are very rare among imprimitive affine groups of rank 3.

The classification narrows down the possibilities for these groups in the context of affine group theory.

Abstract

This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.