Finite groups · Group

$\mathrm{A}_{5}$ Alternating group on five letters

The smallest nonabelian finite simple group, realized by the rotational symmetries of the icosahedron.

Compare this group

Executable coverageSimple groupRotations

Definition

Even permutations of five objects

Definition sources

Order: $60$
Center: Trivial

Reference properties

Property	Value	Evidence
Order	$60$	Sources
Center	Trivial	Sources
Abelian	No	Sources
Simple finite group	Yes	Sources
Ordinary complex irrep dimensions	1, 3, 3, 4, 5	Sources

Representations

Five ordinary complex irreps have dimensions 1, 3, 3, 4 and 5; all have real type. [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
$1$	1	Real
$3$	3	Real
$3'$	3	Real
$4$	4	Real
$5$	5	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

Icosahedral symmetry, finite simple groups, and the relationship to the binary icosahedral spin cover. [sources]

Names and aliases

PSL(2,5) · PSL(2,4) · Icosahedral 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

$\mathrm{A}_{4}$
Alternating group on four letters
Related group
$\mathrm{SO}(3)$
Rotation group in three dimensions
Related group
$\mathrm{A}_{n}$
Simple alternating groups
Alternating simple family
$\mathrm{PSL}(n,q)$
Projective special linear finite simple groups
PSL(2,4) ≅ PSL(2,5)

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 five objects