# What is the name for this generalization of dihedral groups?

A dihedral group $$D$$ can be defined as the group generated by elements $$r$$ and $$s$$, where $$r$$ has order $$n$$, $$s$$ has order $$2$$, and $$sas = a^{-1}$$ for all $$a \in \langle r \rangle$$. It seems that more generally, given a finite abelian group $$G$$, we could define a group $$G \cup sG$$ where $$s$$ has order $$2$$, and $$sas = a^{-1}$$ for all $$a \in G$$. This would essentially taking the maximal cyclic subgroup of a dihedral group and replacing it with a general abelian group. Is there a name for this type of group?

Thanks.