Quasi-dihedral group

A finite $2$-group defined by generators $x,y$ and defining relations

$$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$

where $m\geq4$. The order of a quasi-dihedral group is $2^m$; the group has a cyclic invariant subgroup of index 2. The name was given because of the similarity of the defining relations with those of a dihedron group; however, a quasi-dihedral group is not isomorphic to the latter for any value of $m$. A quasi-dihedral group is sometimes called a semi-dihedral group.

References

[1]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1967)