Quasidihedral group

For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2n which have a cyclic subgroup of index 2.

One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class.

In Bertram Huppert's text Endliche Gruppen, this group is called a "Quasidiedergruppe".

Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article.

All give the same presentation for this group: The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just G or Mm(2).

Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring

The groups of order pn and nilpotency class n − 1 were the beginning of the classification of all p-groups via coclass.

The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose derived subgroup has index 4.