Groups of order 24: Difference between revisions
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! Quantity !! Value | ! Quantity !! Value | ||
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| Total number of groups || 15 | | Total number of groups || [[count::15]] | ||
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| Number of abelian groups || 3 | | Number of abelian groups || 3 | ||
Revision as of 16:34, 15 June 2011
This article gives information about, and links to more details on, groups of order 24
See pages on algebraic structures of order 24 | See pages on groups of a particular order
This article gives basic information comparing and contrasting groups of order 24.
Statistics at a glance
| Quantity | Value |
|---|---|
| Total number of groups | 15 |
| Number of abelian groups | 3 |
| Number of nilpotent groups | 5 |
| Number of solvable groups | 15 |
| Number of simple groups | 0 |
The list
There are 15 groups of order 24.
| Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Derived length |
|---|---|---|---|
| nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | 2 |
| cyclic group:Z24 | 2 | 1 | 1 |
| special linear group:SL(2,3) | 3 | not nilpotent | 3 |
| dicyclic group:Dic24 | 4 | not nilpotent | 2 |
| direct product of S3 and Z4 | 5 | not nilpotent | 2 |
| dihedral group:D24 | 6 | not nilpotent | 2 |
| direct product of Dic12 and Z2 | 7 | not nilpotent | 2 |
| SmallGroup(24,8) | 8 | not nilpotent | 2 |
| direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | 1 |
| direct product of D8 and Z3 | 10 | 2 | 2 |
| direct product of Q8 and Z3 | 11 | 2 | 2 |
| symmetric group:S4 | 12 | not nilpotent | 3 |
| direct product of A4 and Z2 | 13 | not nilpotent | 2 |
| direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | 2 |
| direct product of E8 and Z3 | 15 | 1 | 1 |
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.
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 |