Finite groups · Group

$\mathrm{S}_{3}$ Symmetric group on three letters

The permutation group of three objects, also the full symmetry group of an equilateral triangle.

Compare this group

Executable coveragePermutationsDiscrete symmetry

Definition

⟨r,s | r³ = s² = 1, srs = r⁻¹⟩

Definition sources

Order: $6$
Center: Trivial

Reference properties

Property	Value	Evidence
Order	$6$	Sources
Center	Trivial	Sources
Abelian	No	Sources
Simple finite group	No	Sources
Ordinary complex irrep dimensions	1, 1, 2	Sources

Representations

There are three ordinary complex irreps: the trivial and sign representations, and a real-type doublet. [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
$1$	1	Real
sgn	1	Real
$2$	2	Real

Complete dimension and reality summary for ordinary complex irreps. This does not supply explicit matrices, every character value, or modular representations. Labels follow the conventions stated in this record.

In physics

Permutation symmetry and a compact first example of nonabelian character theory. [sources]

Names and aliases

D3 · D6 · Triangle symmetries

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathrm{S}_{4}$
Symmetric group on four letters
Related group
$\mathrm{A}_{4}$
Alternating group on four letters
Related group
$\mathrm{S}_{n}$
Finite symmetric groups
Symmetric family
$\mathrm{D}_{2n}$
Finite dihedral groups
D₆ in the order-six convention

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

⟨r,s | r³ = s² = 1, srs = r⁻¹⟩