Let $p,q,r$ be three distinct odd primes, and $G$ a finite group with $|G|$ divisible by $p,q,r$ to the first power only. Let $x,y,z \in G$ be of order $p,q,r$ respectively. Assume (a.) $[x,y] = [y,z] = [z, x'] = 1$ where $x' \in G$ is of order $p$; (b.) $xy, yz, zx'$ are all all real elements in $G$; that is there exist $u,v, w \in G$ with $(xy)^u = (xy)^{-1}, (yz)^v = (yz)^{-1}, (zx')^{w} = (zx')^{-1}$.

Question: Do these assumptions guarantee the existence of an element of order $pqr$ in $G$, or if not is there a counter-example?