Finitely presented algebras defined by permutation relations of dihedral type

The class of finitely presented algebras over a field $K$ with a set of generators $a_1,\dots ,a_n$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{\sigma (1)} a_{\sigma (2)} \cdots a_{\sigma (n)}$,where$\sigma$runsthroughasubset$H$ofthesymmetricgroup$$\text{Sym}$_{n}$ofdegree$n$,isinvestigated\mathrm{.}Groups$H$ in which the cyclic group $⟨(1,2,\dots ,n)⟩$ is a normal subgroup of index $2$ are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid $S_n(H)$ with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovski\u{\i}. The universal group $G_n$ of $S_n(H)$ is a unique product group, and it is the central localization of a cancellative subsemigroup of $S_n(H)$. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra $K[S_n(H)]$ is semiprimitive.