Alternating group on six letters
A finite simple group in both the alternating and projective linear classification families.
Reference properties
Representation scope
The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities. [sources]
Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue. [sources]
| Irrep label | Complex dimension | Reality type |
|---|---|---|
| ρ1 | 1 | Real |
| ρ2 | 5 | Real |
| ρ3 | 5 | Real |
| ρ4 | 8 | Real |
| ρ5 | 8 | Real |
| ρ6 | 9 | Real |
| ρ7 | 10 | Real |
Structure and conventions
Both five-dimensional irreps and both eight-dimensional irreps are distinct; dimension alone does not label an irrep. [sources]
Names and aliases
Alt(6) · PSL(2,9) · L2(9) · PSp(4,2) derived subgroup
Related objects and maps
- Family
- Family
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
The group of even permutations of six letters; also PSL(2,9).
- GroupNames: A₆Heading and AliasesOpen source
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
group:a-6#definitionOrdinary complex irreducibles
1, 5, 5, 8, 8, 9, 10
Assumptions: Characteristic zero · Finite-dimensional irreducibles over C · Dimensions listed with multiplicity
- GroupNames: A₆Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:a-6#irrepsGAP SmallGroups ID
SmallGroup(360,118)
- GroupNames: A₆Aliases; GAP/Magma code by IDOpen source
group:a-6#smallgroups-idRepresentation scope
The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.
- GroupNames: A₆Character tableOpen source
group:a-6#representations-1Representation scope
Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.
- GroupNames: A₆Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:a-6#representations-2Structure and conventions
Both five-dimensional irreps and both eight-dimensional irreps are distinct; dimension alone does not label an irrep.
- GroupNames: A₆Heading; Centre; group-property labels; AliasesOpen source
group:a-6#structure-1