Classification of finite 2-groups of maximal class

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This article gives a classification statement for certain kinds of groups of prime power order, subject to additional constraints.
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Statement

Let G be a group of order 2^n and nilpotency class n - 1, where n \ge 4. In other words, G is a 2-group that is also a Maximal class group (?). Then, G has a cyclic maximal subgroup M \cong \mathbb{Z}/2^{n-1}\mathbb{Z}, and it is one of the following groups:

  1. A dihedral group: it is a semidirect product of M and a cyclic group of order two, which acts on M via multiplication by -1.
  2. A semidihedral group: a semidirect product of M and a cyclic group of order two, which acts on M via multiplication by 2^{n-2} - 1.
  3. A generalized quaternion group.

Note that in the case n = 3, we only get the dihedral group:D8 and the quaternion group, and no semidihedral group.

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References

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