On Zappa-Sz\'ep products of two semidihedral groups

Abstract

Let $n, m \ge 4$. We classify the Zappa--Sz\'ep products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong \mathrm{SD}_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong \mathrm{SD}_{2^m}$, according to the cores of $\langle x\rangle$ and $\langle z\rangle$ in~$G$. First, when both $\langle x\rangle$ and $\langle z\rangle$ are normal in~$G$, we obtain a complete classification of such exact products by an explicit system of six polynomial congruences. Second, when the cores $\langle x\rangle^G$ and $\langle z\rangle^G$ are arbitrary subgroups of $\langle x\rangle$ and $\langle z\rangle$, under the simplifying assumption $[x, z] = 1$ we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu--Yu classification~\cite{HuYu2025} for dihedral groups. In contrast with the dihedral case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, showing that the parameter space in the semidihedral setting is strictly richer than its dihedral analogue.