Finite groups · Group

$\mathrm{S}_{4}$ Symmetric group on four letters

The permutation group of four objects. It is also isomorphic to the orientation-preserving rotation group of a cube.

Compare this group

Executable coverageFlavorRotations

Definition

Permutations of four objects

Definition sources

Order: $24$
Center: Trivial

Reference properties

Property	Value	Evidence
Order	$24$	Sources
Center	Trivial	Sources
Abelian	No	Sources
Simple finite group	No	Sources
Ordinary complex irrep dimensions	1, 1, 2, 3, 3	Sources

Representations

Five ordinary complex irreps have dimensions 1, 1, 2, 3 and 3; each has real type. [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
$1$	1	Real
sgn	1	Real
$2$	2	Real
$3$	3	Real
$3'$	3	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

Discrete-flavor constructions, permutation symmetries, and cubic rotations with a specified action. [sources]

Names and aliases

Octahedral rotation group

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

Related objects and maps

$\mathrm{S}_{3}$
Symmetric group on three letters
Related group
$\mathrm{A}_{4}$
Alternating group on four letters
Related group
$\mathrm{SO}(3)$
Rotation group in three dimensions
Related group
$\mathrm{S}_{n}$
Finite symmetric groups
Symmetric 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

Permutations of four objects