Subgroup structure of groups of order 24: Difference between revisions

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information type = subgroup structure|
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connective = of}}
==Sylow subgroups==
===2-Sylow subgroups===
Here is the occurrence summary:
<section begin="2-Sylow summary"/>
{| class="sortable" border="1"
! Group of order 8 !! GAP ID (second part) !! Number of groups of order 24 in which it is a 2-Sylow subgroup !! List of these groups !! Second part of GAP IDs of these groups
|-
| [[cyclic group:Z8]] || 1 || 2 || [[nontrivial semidirect product of Z3 and Z8]], [[cyclic group:Z24]] || 1, 2
|-
| [[direct product of Z4 and Z2]] || 2 || 3 || [[direct product of S3 and Z4]], [[direct product of Dic12 and Z2]], [[direct product of Z6 and Z4]] || 5, 7, 9
|-
| [[dihedral group:D8]] || 3 || 4 || [[dihedral group:D24]], [[SmallGroup(24,8)]], [[direct product of D8 and Z3]], [[symmetric group:S4]] || 6, 8, 10, 12
|-
| [[quaternion group]] || 4 || 3 || [[special linear group:SL(2,3)]], [[dicyclic group:Dic24]], [[direct product of Q8 and Z3]] || 3, 4, 11
|-
| [[elementary abelian group:E8]] || 5 || 3 || [[direct product of A4 and Z2]], [[direct product of D12 and Z2]], [[direct product of E8 and Z3]] || 13, 14, 15
|}
<section end="2-Sylow summary"/>
Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a [[normal Sylow subgroup]] for the prime 2. The latter happens if and only if we have a [[self-normalizing Sylow subgroup]] for the prime 2.
{| class="sortable" border="1"
! Group !! Second part of GAP ID (ID is (24,second part)) !! 2-Sylow subgroup !!  Second part of GAP ID !! Number of 2-Sylow subgroups
|-
| [[nontrivial semidirect product of Z3 and Z8]] || 1 || [[cyclic group:Z8]] || 1 || 3
|-
| [[cyclic group:Z24]] || 2 || [[cyclic group:Z8]] || 1 || 1
|-
| [[special linear group:SL(2,3)]] || 3 || [[quaternion group]] || 4 || 1
|-
| [[dicyclic group:Dic24]] || 4 || [[quaternion group]] || 4 || 3
|-
| [[direct product of S3 and Z4]] || 5 || [[direct product of Z4 and Z2]] || 2 || 3
|-
| [[dihedral group:D24]] || 6 || [[dihedral group:D8]] || 3 || 3
|-
| [[direct product of Dic12 and Z2]] || 7 || [[direct product of Z4 and Z2]] || 2 || 3
|-
| [[SmallGroup(24,8)]] || 8 || [[dihedral group:D8]] || 3 || 3
|-
| [[direct product of Z6 and Z4]] (also, direct product of Z12 and Z2) || 9 || [[direct product of Z4 and Z2]] || 2 || 1
|-
| [[direct product of D8 and Z3]] || 10 || [[dihedral group:D8]] || 3 || 1
|-
| [[direct product of Q8 and Z3]] || 11 || [[quaternion group]] || 4 || 1
|-
| [[symmetric group:S4]] || 12 || [[dihedral group:D8]] || 3 || 3
|-
| [[direct product of A4 and Z2]] || 13 || [[elementary abelian group:E8]] || 5 || 1
|-
| [[direct product of D12 and Z2]] (also direct product of S3 and V4) || 14 || [[elementary abelian group:E8]] || 5 || 3
|-
| [[direct product of E8 and Z3]] || 15 || [[elementary abelian group:E8]] || 5 || 1
|}
===3-Sylow subgroups===
Note that the 3-Sylow subgroup is isomorphic to [[cyclic group:Z3]] in all cases. By the [[congruence condition on Sylow numbers]] as well as the divisibility condition on Sylow numbers, the only possibilities for the number of 3-Sylow subgroups is 1 or 4. In the former case, we have a [[normal Sylow subgroup]]. In the latter case, the normalizer of the Sylow subgroup has order 6, and is thus either [[cyclic group:Z6]] or [[symmetric group:S3]].
{| class="sortable" border="1"
! Group !! Second part of GAP ID (ID is (24,second part)) !! Number of 3-Sylow subgroups !! Normalizer of Sylow subgroup
|-
| [[nontrivial semidirect product of Z3 and Z8]] || 1 || 1 || whole group
|-
| [[cyclic group:Z24]] || 2 || 1 || whole group
|-
| [[special linear group:SL(2,3)]] || 3 || 4 || [[cyclic group:Z6]]
|-
| [[dicyclic group:Dic24]] || 4 || 1 || whole group
|-
| [[direct product of S3 and Z4]] || 5 || 1 || whole group
|-
| [[dihedral group:D24]] || 6 || 1 || whole group
|-
| [[direct product of Dic12 and Z2]] || 7 || 1 || whole group
|-
| [[SmallGroup(24,8)]] || 8 || 1 || whole group
|-
| [[direct product of Z6 and Z4]] (also, direct product of Z12 and Z2) || 9 || 1 || whole group
|-
| [[direct product of D8 and Z3]] || 10 || 1 || whole group
|-
| [[direct product of Q8 and Z3]] || 11 || 1 || whole group
|-
| [[symmetric group:S4]] || 12 || 4 || [[symmetric group:S3]]
|-
| [[direct product of A4 and Z2]] || 13 || 4 || [[cyclic group:Z6]]
|-
| [[direct product of D12 and Z2]] (also direct product of S3 and V4) || 14 || 1 || whole group
|-
| [[direct product of E8 and Z3]] || 15 || 1 || whole group
|}

Revision as of 20:08, 15 July 2011

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

Sylow subgroups

2-Sylow subgroups

Here is the occurrence summary:

Group of order 8 GAP ID (second part) Number of groups of order 24 in which it is a 2-Sylow subgroup List of these groups Second part of GAP IDs of these groups
cyclic group:Z8 1 2 nontrivial semidirect product of Z3 and Z8, cyclic group:Z24 1, 2
direct product of Z4 and Z2 2 3 direct product of S3 and Z4, direct product of Dic12 and Z2, direct product of Z6 and Z4 5, 7, 9
dihedral group:D8 3 4 dihedral group:D24, SmallGroup(24,8), direct product of D8 and Z3, symmetric group:S4 6, 8, 10, 12
quaternion group 4 3 special linear group:SL(2,3), dicyclic group:Dic24, direct product of Q8 and Z3 3, 4, 11
elementary abelian group:E8 5 3 direct product of A4 and Z2, direct product of D12 and Z2, direct product of E8 and Z3 13, 14, 15

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

Group Second part of GAP ID (ID is (24,second part)) 2-Sylow subgroup Second part of GAP ID Number of 2-Sylow subgroups
nontrivial semidirect product of Z3 and Z8 1 cyclic group:Z8 1 3
cyclic group:Z24 2 cyclic group:Z8 1 1
special linear group:SL(2,3) 3 quaternion group 4 1
dicyclic group:Dic24 4 quaternion group 4 3
direct product of S3 and Z4 5 direct product of Z4 and Z2 2 3
dihedral group:D24 6 dihedral group:D8 3 3
direct product of Dic12 and Z2 7 direct product of Z4 and Z2 2 3
SmallGroup(24,8) 8 dihedral group:D8 3 3
direct product of Z6 and Z4 (also, direct product of Z12 and Z2) 9 direct product of Z4 and Z2 2 1
direct product of D8 and Z3 10 dihedral group:D8 3 1
direct product of Q8 and Z3 11 quaternion group 4 1
symmetric group:S4 12 dihedral group:D8 3 3
direct product of A4 and Z2 13 elementary abelian group:E8 5 1
direct product of D12 and Z2 (also direct product of S3 and V4) 14 elementary abelian group:E8 5 3
direct product of E8 and Z3 15 elementary abelian group:E8 5 1

3-Sylow subgroups

Note that the 3-Sylow subgroup is isomorphic to cyclic group:Z3 in all cases. By the congruence condition on Sylow numbers as well as the divisibility condition on Sylow numbers, the only possibilities for the number of 3-Sylow subgroups is 1 or 4. In the former case, we have a normal Sylow subgroup. In the latter case, the normalizer of the Sylow subgroup has order 6, and is thus either cyclic group:Z6 or symmetric group:S3.

Group Second part of GAP ID (ID is (24,second part)) Number of 3-Sylow subgroups Normalizer of Sylow subgroup
nontrivial semidirect product of Z3 and Z8 1 1 whole group
cyclic group:Z24 2 1 whole group
special linear group:SL(2,3) 3 4 cyclic group:Z6
dicyclic group:Dic24 4 1 whole group
direct product of S3 and Z4 5 1 whole group
dihedral group:D24 6 1 whole group
direct product of Dic12 and Z2 7 1 whole group
SmallGroup(24,8) 8 1 whole group
direct product of Z6 and Z4 (also, direct product of Z12 and Z2) 9 1 whole group
direct product of D8 and Z3 10 1 whole group
direct product of Q8 and Z3 11 1 whole group
symmetric group:S4 12 4 symmetric group:S3
direct product of A4 and Z2 13 4 cyclic group:Z6
direct product of D12 and Z2 (also direct product of S3 and V4) 14 1 whole group
direct product of E8 and Z3 15 1 whole group
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