Classification of groups of order 24

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Contents

This article gives specific information, namely, classification, about a family of groups, namely: groups of order 24.
View classification of group families | View classification of groups of a particular order |View other specific information about groups of order 24

Statement

There are, up to isomorphism, 15 groups of order 24, given as follows:

Group Second part of GAP ID (ID is (24,second part)) Nilpotency class Derived length 2-Sylow subgroup Is the 2-Sylow subgroup normal? Is the 3-Sylow subgroup normal?
nontrivial semidirect product of Z3 and Z8 1 not nilpotent 2 cyclic group:Z8 No Yes
cyclic group:Z24 2 1 1 cyclic group:Z8 Yes Yes
special linear group:SL(2,3) 3 not nilpotent 3 quaternion group Yes No
dicyclic group:Dic24 4 not nilpotent 2 quaternion group No Yes
direct product of S3 and Z4 5 not nilpotent 2 direct product of Z4 and Z2 No Yes
dihedral group:D24 6 not nilpotent 2 dihedral group:D8 No Yes
direct product of Dic12 and Z2 7 not nilpotent 2 direct product of Z4 and Z2 No Yes
SmallGroup(24,8) 8 not nilpotent 2 dihedral group:D8 No Yes
direct product of Z6 and Z4 (also, direct product of Z12 and Z2) 9 1 1 direct product of Z4 and Z2 Yes Yes
direct product of D8 and Z3 10 2 2 dihedral group:D8 Yes Yes
direct product of Q8 and Z3 11 2 2 quaternion group Yes Yes
symmetric group:S4 12 not nilpotent 3 dihedral group:D8 No No
direct product of A4 and Z2 13 not nilpotent 2 elementary abelian group:E8 Yes No
direct product of D12 and Z2 (also direct product of S3 and V4) 14 not nilpotent 2 elementary abelian group:E8 No Yes
direct product of E8 and Z3 15 1 1 elementary abelian group:E8 Yes Yes