Finite imprimitive rank 3 affine groups

Theorem

Let $G⩽\mathrm{Sym}(\Omega )$ be a finite semiprimitive permutation group of rank 3 with point stabiliser $H$ and assume that $G$ is not innately transitive. Then up to permutation isomorphism, one of the following holds:

• (a) $(G,H)=\left(3.S_6,S_5\right)$ or $\left(2.{\mathrm{M}}_{12},{\mathrm{M}}_{11}\right)$.
• (b) $\Omega$ is the set of $C$-orbits on ${\mathbb{F}}_q^d\backslash \{0\}$ for ${\mathbb{Z}}_{(q-1)/r}\cong C<Z\left({\mathrm{GL}}_d(q)\right)$, and we have

$$r\cdot {\mathrm{PSL}}_d(q)\cong {\mathrm{SL}}_d(q)C/C⩽G⩽\Gamma {\mathrm{L}}_d(q)/C\cong r\cdot {\mathrm{P}}_d{\mathrm{L}}_d(q)$$

where $(d,q)\ne (2,3),q=p^f,p$ is a prime and $r$ is a primitive prime divisor of $p^{r-1}-1$.

• (c) $G$ has a regular normal subgroup $N$, where $N$ is a special p-group for some prime $p$.

Theorem

Let $G⩽\mathrm{Sym}(\Omega )$ be a finite transitive rank 3 permutation group with nontrivial block system $\mathcal{B}$. Let $B,B^{\prime }\in \mathcal{B}$ be distinct blocks and assume $G_B^B$ is affine. Then one of the following holds:

• (A) $G$ is either innately transitive, or it is permutation isomorphic to a group recorded in case (a) or (b) of ^e8abd5;
• (B) $N⩽G⩽N:\mathrm{Aut}(N)$, where $N$ is a regular normal subgroup and $\mathrm{Aut}(N)$ has at most 3 orbits on $N$;
• (C) $K_{(B)}$ is transitive on $B^{\prime }$;
• (D) $K_{(B)}\ne 1$ is intransitive on $B^{\prime }$, and $G$ has an elementary abelian self-centralising normal subgroup.