Finite groups · Group

$2O$ Binary octahedral group

The binary rotation group of the cube or octahedron, a central double cover of S₄.

Compare this group

Reference coverageFinite referenceNonabelian

Definition

The inverse image of the octahedral rotation group in SU(2).

Definition sources

Order: $48$
Center: $\mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Order	$48$	Sources
Center	$\mathrm{C}_{2}$	Sources
Abelian	No	Sources
Simple	No	Sources
Ordinary complex irreducibles	1, 1, 2, 2, 2, 3, 3, 4	Sources
GAP SmallGroups ID	SmallGroup(48,28)	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	1	Real
ρ3	2	Real
ρ4	2	Quaternionic
ρ5	2	Quaternionic
ρ6	3	Real
ρ7	3	Real
ρ8	4	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 faithful doublets and the four-dimensional irrep are quaternionic. The other irreps descend to the rotational quotient S₄. [sources]

Names and aliases

CSU(2,3) · Binary octahedral · 2O

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
$\mathrm{S}_{4}$
Symmetric group on four letters
2O / C₂ ≅ S₄
$\mathrm{SU}(2)$
Special unitary group
Binary octahedral 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 octahedral rotation group in SU(2).

Character Theory of Finite Groups: Day 4 — The McKay CorrespondenceBinary polyhedral groupsOpen source
GroupNames: CSU(2,3), binary octahedralAliasesOpen source

group:binary-octahedral#definition

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Order

48

GroupNames: CSU(2,3), binary octahedralHeading: orderOpen source

group:binary-octahedral#order

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Center

C₂

GroupNames: CSU(2,3), binary octahedralCentreOpen source

group:binary-octahedral#center

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Abelian

No

GroupNames: CSU(2,3), binary octahedralCommutator; character tableOpen source

group:binary-octahedral#abelian

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Simple

No

GroupNames: CSU(2,3), binary octahedralChief series; CentreOpen source

group:binary-octahedral#simple

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Ordinary complex irreducibles

1, 1, 2, 2, 2, 3, 3, 4

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

GroupNames: CSU(2,3), binary octahedralCharacter table: identity column and real/complex/symplectic annotationsOpen source
GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source

group:binary-octahedral#irreps

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GAP SmallGroups ID

SmallGroup(48,28)

GroupNames: CSU(2,3), binary octahedralAliases; GAP/Magma code by IDOpen source

group:binary-octahedral#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: CSU(2,3), binary octahedralCharacter tableOpen source

group:binary-octahedral#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: CSU(2,3), binary octahedralCharacter table: identity column and real/complex/symplectic annotationsOpen source
GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source

group:binary-octahedral#representations-2

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Structure and conventions

The faithful doublets and the four-dimensional irrep are quaternionic. The other irreps descend to the rotational quotient S₄.

GroupNames: CSU(2,3), binary octahedralHeading; Centre; group-property labels; AliasesOpen source

group:binary-octahedral#structure-1

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Family

family:binary-polyhedral

GroupNames: CSU(2,3), binary octahedralHeading; AliasesOpen source

group:binary-octahedral#relation-1

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2O / C₂ ≅ S₄

group:s-4

GroupNames: CSU(2,3), binary octahedralCentre: G/ZOpen source

group:binary-octahedral#relation-2

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Binary octahedral embedding

group:su-2

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

group:binary-octahedral#relation-3

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