See symmetric group:S3. We take the symmetric group on the set of size three.
See element structure of symmetric group:S3 for full details.
Element orders and conjugacy class structureReview the conjugacy class structure: [SHOW MORE]
Multiplication, conjugacy and generating setsReview the multiplication table in cycle decomposition notation: [SHOW MORE]
Review the multiplication table in one-line notation: [SHOW MORE]
Conjugation and commutator operationsReview the conjugation operation: [SHOW MORE]
Review the commutator operation: [SHOW MORE]
See subgroup structure of symmetric group:S3 for background information.
Basic stuffSummary table on the structure of subgroups: [SHOW MORE]
View the lattice of subgroups as a picture: [SHOW MORE]
For background, see linear representation theory of symmetric group:S3.View overall summary: [SHOW MORE]
View representations summary: [SHOW MORE]
View character table: [SHOW MORE]
Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the one-dimensional trivial representation, the one-dimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a two-dimensional representation.