Dicyclic group of order twelve
A central double cover of S₃ with one real and one quaternionic doublet.
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 | Complex |
| ρ4 | 1 | Complex |
| ρ5 | 2 | Real |
| ρ6 | 2 | Quaternionic |
Structure and conventions
Its abelianization is C₄; two singlets form a complex-conjugate pair. [sources]
Names and aliases
Dic(3) · C3 ⋊ C4 · Binary dihedral group of order 12
Related objects and maps
- Family
- Dic₃ / 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
The group with a⁶ = 1, x² = a³ and xax⁻¹ = a⁻¹.
- GroupNames: Dic₃Heading; Aliases; Generators and relationsOpen source
group:dic-3#definitionOrdinary complex irreducibles
1, 1, 1, 1, 2, 2
Assumptions: Characteristic zero · Finite-dimensional irreducibles over C · Dimensions listed with multiplicity
- GroupNames: Dic₃Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:dic-3#irrepsGAP SmallGroups ID
SmallGroup(12,1)
- GroupNames: Dic₃Aliases; GAP/Magma code by IDOpen source
group:dic-3#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: Dic₃Character tableOpen source
group:dic-3#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: Dic₃Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:dic-3#representations-2Structure and conventions
Its abelianization is C₄; two singlets form a complex-conjugate pair.
- GroupNames: Dic₃Heading; Centre; group-property labels; AliasesOpen source
group:dic-3#structure-1