Classification of finite 2-groups of maximal class
From Groupprops
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 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.
Note that in the case , we only get the dihedral group:D8 and the quaternion group, and no semidihedral group.
Particular cases
(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 |
Related facts
- Finite 2-groups of same order and maximal class are isoclinic
- Classification of finite p-groups with cyclic maximal subgroup
- Classification of finite 3-groups of maximal class
- Finite non-abelian 2-group has maximal class iff its abelianization has order four
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 194, Section 5.4 (p-groups of small depth), Theorem 4.5, ^{More info}