Finite groups with large Chebotarev invariant

A subset $\{g_1,\dots ,g_d\}$ of a finite group $G$ is said to invariably generate $G$ if the set $\{g_1^{x_1},\dots ,g_d^{x_d}\}$ generates $G$ for every choice of $x_i\in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The authors recently showed that for each $ϵ>0$, there exists a constant $c_{ϵ}$ such that $C(G)\le (1+ϵ)\sqrt{\mathrm{\mid }G\mathrm{\mid }}+c_{ϵ}$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each $\alpha >0$ there exists an absolute constant ${\delta }_{\alpha }$ such that if $G$ is a finite group and $C(G)>\alpha \sqrt{\mathrm{\mid }G\mathrm{\mid }}$, then $G$ has a section $X\mathrm{/}Y$ such that $|X/Y|\geq \delta_{\alpha}\sqrt{|G|}$,and$X/Y\cong $\mathbb{F}$_q\rtimes H$ for some prime power $q$, with $H\le {\mathbb{F}}_q^{\times }$.