On Zappa-Szép products of two semidihedral groups

Abstract

Let $n,m\ge 4$. We classify the Zappa--Szép products $G=HK$ with $H=⟨x⟩⋊⟨y⟩\cong {\mathrm{SD}}_{2^n}$ and $K=⟨z⟩⋊⟨w⟩\cong {\mathrm{SD}}_{2^m}$, according to the cores of $⟨x⟩$ and $⟨z⟩$ in~$G$. First, when both $⟨x⟩$ and $⟨z⟩$ 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 $⟨x{⟩}^G$ and $⟨z{⟩}^G$ are arbitrary subgroups of $⟨x⟩$ and $⟨z⟩$, 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.