Endomorphism structure of groups of order 12
This article gives specific information, namely, endomorphism structure, about a family of groups, namely: groups of order 12.
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This page discusses the endomorphism structure of groups of order 12.
The groups are:
The list
| Group | Second part of GAP ID (GAP ID is (12,second part)) | Abelian? | 2-Sylow subgroup | Is the 2-Sylow subgroup normal? | Is the 3-Sylow subgroup normal? |
|---|---|---|---|---|---|
| dicyclic group:Dic12 | 1 | No | cyclic group:Z4 | No | Yes |
| cyclic group:Z12 | 2 | Yes | cyclic group:Z4 | Yes | Yes |
| alternating group:A4 | 3 | No | Klein four-group | Yes | No |
| dihedral group:D12 | 4 | No | Klein four-group | No | Yes |
| direct product of Z6 and Z2 | 5 | Yes | Klein four-group | Yes | Yes |
Automorphism group
The automorphism groups of the groups are as follows:
| Group | Second part of GAP ID | Isomorphism class of automorphism group | Order of automorphism group |
|---|---|---|---|
| dicyclic group:Dic12 | 1 | dihedral group:D12 | 12 |
| cyclic group:Z12 | 2 | Klein four-group | 4 |
| alternating group:A4 | 3 | symmetric group:S4 | 24 |
| dihedral group:D12 | 4 | dihedral group:D12 | 12 |
| direct product of Z6 and Z2 | 5 | dihedral group:D12 | 12 |