Finite groups · Group

$2I$ Binary icosahedral group

A perfect central double cover of A₅; it is quasisimple, with a nontrivial center.

Compare this group

Reference coverageFinite referenceNonabelian

Definition

The inverse image of the icosahedral rotation group in SU(2); abstractly SL(2,5).

Definition sources

Order: $120$
Center: $\mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Order	$120$	Sources
Center	$\mathrm{C}_{2}$	Sources
Abelian	No	Sources
Simple	No	Sources
Ordinary complex irreducibles	1, 2, 2, 3, 3, 4, 4, 5, 6	Sources
GAP SmallGroups ID	SmallGroup(120,5)	Sources

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]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
ρ1	1	Real
ρ2	2	Quaternionic
ρ3	2	Quaternionic
ρ4	3	Real
ρ5	3	Real
ρ6	4	Real
ρ7	4	Quaternionic
ρ8	5	Real
ρ9	6	Quaternionic

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.

Structure and conventions

The two four-dimensional irreps differ: one is real and descends to A₅; the other is quaternionic and faithful. [sources]

Names and aliases

SL(2,5) · 2.A5 · Binary icosahedral

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

Related objects and maps

$2T,\quad 2O,\quad 2I$
Exceptional binary polyhedral groups
Family
$\mathrm{A}_{5}$
Alternating group on five letters
2I / C₂ ≅ A₅
$\mathrm{SU}(2)$
Special unitary group
Binary icosahedral embedding

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 inverse image of the icosahedral rotation group in SU(2); abstractly SL(2,5).

Character Theory of Finite Groups: Day 4 — The McKay CorrespondenceBinary polyhedral groupsOpen source
GroupNames: SL(2,5), binary icosahedralAliasesOpen source

group:binary-icosahedral#definition

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Order

120

GroupNames: SL(2,5), binary icosahedralHeading: orderOpen source

group:binary-icosahedral#order

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Center

C₂

GroupNames: SL(2,5), binary icosahedralCentreOpen source

group:binary-icosahedral#center

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Abelian

No

GroupNames: SL(2,5), binary icosahedralCommutator; character tableOpen source

group:binary-icosahedral#abelian

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Simple

No

GroupNames: SL(2,5), binary icosahedralChief series; CentreOpen source

group:binary-icosahedral#simple

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Ordinary complex irreducibles

1, 2, 2, 3, 3, 4, 4, 5, 6

Assumptions: Characteristic zero · Finite-dimensional irreducibles over C · Dimensions listed with multiplicity

GroupNames: SL(2,5), binary icosahedralCharacter table: identity column and real/complex/symplectic annotationsOpen source
GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source

group:binary-icosahedral#irreps

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GAP SmallGroups ID

SmallGroup(120,5)

GroupNames: SL(2,5), binary icosahedralAliases; GAP/Magma code by IDOpen source

group:binary-icosahedral#smallgroups-id

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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.

GroupNames: SL(2,5), binary icosahedralCharacter tableOpen source

group:binary-icosahedral#representations-1

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Representation 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: SL(2,5), binary icosahedralCharacter table: identity column and real/complex/symplectic annotationsOpen source
GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source

group:binary-icosahedral#representations-2

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Structure and conventions

The two four-dimensional irreps differ: one is real and descends to A₅; the other is quaternionic and faithful.

GroupNames: SL(2,5), binary icosahedralHeading; Centre; group-property labels; AliasesOpen source

group:binary-icosahedral#structure-1

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Family

family:binary-polyhedral

GroupNames: SL(2,5), binary icosahedralHeading; AliasesOpen source

group:binary-icosahedral#relation-1

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2I / C₂ ≅ A₅

group:a-5

GroupNames: SL(2,5), binary icosahedralCentre: G/ZOpen source

group:binary-icosahedral#relation-2

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Binary icosahedral embedding

group:su-2

Character Theory of Finite Groups: Day 4 — The McKay CorrespondenceBinary polyhedral groupsOpen source

group:binary-icosahedral#relation-3

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