Let $G=P \ltimes Q$ be a finite non-nilpotent group, where $P \in {\rm Syl}_q(G)$ is non-cyclic $p$-subgroup and $Q \in {\rm Syl}_q(G)$ is non-abelian. Suppose that $\bar{Q}:=Q/ \Phi(Q) \cong \Bbb{Z}_q \times \Bbb{Z}_q$ and $\bar{Q}$ is an irreducible $P$-module and $\Phi(Q)$ is minimal normal subgroup of $G$. I want to know why $Q^{\prime}/\Gamma_3(Q)$ is cyclic? Also i want conclude that $|\Phi(Q)|=q$.
MY work: I know since $Q$ is nilpotent $\exists n \in \Bbb{N} \ ;\ \Gamma_n(Q)=1$. Also we have $\Gamma_1(Q)=Q$ and $\Gamma_2(Q)=Q^{\prime}$ and $Q^{\prime}/\Gamma_3(Q) \leq Z(Q/\Gamma_3(Q))$.
Also i know $Q^{\prime} \leq \Phi(Q)$(Because $Q$ is nilpotent), and since $Q^{\prime} \trianglelefteq G$, we get $Q^{\prime}=\Phi(Q)$.