  For a regular representation $H \subseteq \text{Sym}_n$ of the generalized
quaternion group of order $n=4k$, with $k\geq 2$, the monoid $S_n(H)$ presented
with generators $a_1,a_2,\dots ,a_n$ and with relations $a_1a_2\cdots
a_n=a_{\sigma(1)}a_{\sigma(2)}\cdots a_{\sigma(n)}$, for all $\sigma\in H$, is
investigated. It is shown that $S_n(H)$ has the two unique product property. As
a consequence, for any field $K$, the monoid algebra $K[S_n(H)]$ is a domain
with trivial units which is semiprimitive.
