Finite groups · Group

$\mathrm{A}_{4}$ Alternating group on four letters

The even permutations of four objects, realized as the orientation-preserving rotational symmetry of a tetrahedron.

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Executable coverageFlavorInvariants

Definition

Even permutations of four objects

Definition sources

Order: $12$
Center: Trivial

Reference properties

Property	Value	Evidence
Order	$12$	Sources
Center	Trivial	Sources
Abelian	No	Sources
Simple finite group	No	Sources
Ordinary complex irrep dimensions	1, 1, 1, 3	Sources

Representations

The irreps are 1, 1′, 1″ and 3. The two nontrivial singlets are complex conjugates; the triplet has real type. [sources]

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

Ordinary character table

Class	1	2A	3A	3B
Class size	1	3	4	4
$1$	$1$	$1$	$1$	$1$
$1'$	$1$	$1$	$\omega$	$\omega^2$
$1''$	$1$	$1$	$\omega^2$	$\omega$
$3$	$3$	−1	$0$	$0$

ω = exp(2πi/3). Classes 3A and 3B fix the singlet labeling. Character-table source

In physics

Discrete-flavor models and an exact example of invariant counting with field statistics. [sources]

Names and aliases

Tetrahedral 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

$\mathrm{SO}(3)$
Rotation group in three dimensions
Related group
$\mathrm{S}_{4}$
Symmetric group on four letters
Related group
$\mathrm{C}_{3}$
Cyclic group of order three
Related group

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

Even permutations of four objects