Classification of finite 2-groups of maximal class
This article gives a classification statement for certain kinds of groups of prime power order, subject to additional constraints.
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Let be a group of order and nilpotency class , where . In other words, is a 2-group that is also a maximal class group. Then, has a cyclic maximal subgroup , and it is one of the following groups:
- A dihedral group: it is a semidirect product of and a cyclic group of order two, which acts on via multiplication by -1.
- A semidihedral group: a semidirect product of and a cyclic group of order two, which acts on via multiplication by .
- A generalized quaternion group.
|(equals nilpotency class)||dihedral group of order (this is the only capable group among the three)||semidihedral group of order||generalized quaternion group of order|
|3||2||8||dihedral group:D8||--||quaternion group|
|4||3||16||dihedral group:D16||semidihedral group:SD16||generalized quaternion group:Q16|
|5||4||32||dihedral group:D32||semidihedral group:SD32||generalized quaternion group:Q32|
|6||5||64||dihedral group:D64||semidihedral group:SD64||generalized quaternion group:Q64|