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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.

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