Throughout studying a paper about finite $p$-groups, I have the following question
Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote the i'th term of the lower central series of $G$. Let $\gamma_3(G)$ has prime order and $\displaystyle\frac{G}{\gamma_3(G)}\cong M(p^n), n\geq 3$, where $M(p^n)$ denotes the modular $p$-group with the following presentation $$\langle a, b| a^{p^{n-1}}=b^{p}=1, [a,b]=a^{p^{n-2}}\rangle,$$ where $p$ is a prime and $n\geq 3$ if $p$ is odd, $n\geq 4$ otherwise. Obviously $M(p^n)$ has a cyclic subgroup of index $p$ (say $\langle a\rangle$).
Is it true that $G$ has a cyclic subgroup of index $p$?
Any answer will be appreciated!