Dihedral group of order twelve
Hexagon symmetries; abstractly C₂ × S₃, with entirely real ordinary irreducibles.
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 | 1 | Real |
| ρ3 | 1 | Real |
| ρ4 | 1 | Real |
| ρ5 | 2 | Real |
| ρ6 | 2 | Real |
Structure and conventions
The central half-turn generates C₂. Quotienting by the center gives S₃. The cited page calls this group D₆. [sources]
Names and aliases
Dihedral(12) · Dih(6) · D6 (polygon convention) · C2 × S3
Related objects and maps
- Family
- D₁₂ / C₂ ≅ S₃
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
Symmetries of the regular hexagon: r⁶ = s² = 1 and srs⁻¹ = r⁻¹.
- GroupNames: D₆, order 12Heading; Aliases; Generators and relationsOpen source
group:d-12#definitionCenter
C₂
group:d-12#centerOrdinary complex irreducibles
1, 1, 1, 1, 2, 2
Assumptions: Characteristic zero · Finite-dimensional irreducibles over C · Dimensions listed with multiplicity
- GroupNames: D₆, order 12Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:d-12#irrepsGAP SmallGroups ID
SmallGroup(12,4)
- GroupNames: D₆, order 12Aliases; GAP/Magma code by IDOpen source
group:d-12#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: D₆, order 12Character tableOpen source
group:d-12#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: D₆, order 12Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:d-12#representations-2Structure and conventions
The central half-turn generates C₂. Quotienting by the center gives S₃. The cited page calls this group D₆.
- GroupNames: D₆, order 12Heading; Centre; group-property labels; AliasesOpen source
group:d-12#structure-1Family
family:finite-dihedral
- GroupNames: D₆, order 12Heading; AliasesOpen source
group:d-12#relation-1