Groups of order 6
This article gives information about, and links to more details on, groups of order 6
See pages on algebraic structures of order 6 | See pages on groups of a particular order
This article gives basic information comparing and contrasting groups of order 6. See also more detailed information on specific subtopics through the links:
| Information type | Page summarizing information for groups of order 6 |
|---|---|
| element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 6 |
| subgroup structure | subgroup structure of groups of order 6 |
| linear representation theory | linear representation theory of groups of order 6 projective representation theory of groups of order 6 modular representation theory of groups of order 6 |
| endomorphism structure, automorphism structure | endomorphism structure of groups of order 6 |
| group cohomology | group cohomology of groups of order 6 |
The list
There are, up to isomorphism, two groups of order 6, indicated in the table below:
| Group | GAP ID (second part) | Abelian? |
|---|---|---|
| symmetric group:S3 | 1 | No |
| cyclic group:Z6 | 2 | Yes |
There are many ways of demonstrating that there are only two groups of order six, including simply looking at the possible multiplication tables. One of the more efficient approaches is via the classification of groups of an order two times a prime, or more generally the classification of groups of order a product of two distinct primes. Since and , the number falls in the two isomorphism classes case in that classification.
Specific information
| Information type | For symmetric group:S3 | For cyclic group:Z6 |
|---|---|---|
| element structure | element structure of symmetric group:S3 | element structure of cyclic group:Z6 |
| subgroup structure | subgroup structure of symmetric group:S3 | subgroup structure of cyclic group:Z6 |
| linear representation theory | linear representation theory of symmetric group:S3 | linear representation theory of cyclic group:Z6 |
| endomorphism structure | endomorphism structure of symmetric group:S3 | endomorphism structure of cyclic group:Z6 |