Finite groups · Group

$2T$ Binary tetrahedral group

The central double cover of A₄, with three doublets of different Frobenius–Schur types.

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Reference coverageFinite referenceNonabelian

Definition

The inverse image of the tetrahedral rotation group in SU(2); abstractly SL(2,3).

Definition sources

Order: $24$
Center: $\mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Order	$24$	Sources
Center	$\mathrm{C}_{2}$	Sources
Abelian	No	Sources
Simple	No	Sources
Ordinary complex irreducibles	1, 1, 1, 2, 2, 2, 3	Sources
GAP SmallGroups ID	SmallGroup(24,3)	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	Complex
ρ3	1	Complex
ρ4	2	Quaternionic
ρ5	2	Complex
ρ6	2	Complex
ρ7	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.

Structure and conventions

The center acts trivially on the three singlets and the triplet; those representations descend to A₄. [sources]

Names and aliases

SL(2,3) · 2.A4 · Q8 ⋊ C3

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
$\mathrm{A}_{4}$
Alternating group on four letters
2T / C₂ ≅ A₄
$\mathrm{SU}(2)$
Special unitary group
Binary tetrahedral 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 tetrahedral rotation group in SU(2); abstractly SL(2,3).

Character Theory of Finite Groups: Day 4 — The McKay CorrespondenceBinary polyhedral groups; binary tetrahedral exampleOpen source
GroupNames: SL(2,3), binary tetrahedralAliasesOpen source

group:binary-tetrahedral#definition

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Order

24

GroupNames: SL(2,3), binary tetrahedralHeading: orderOpen source

group:binary-tetrahedral#order

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Center

C₂

GroupNames: SL(2,3), binary tetrahedralCentreOpen source

group:binary-tetrahedral#center

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Abelian

No

GroupNames: SL(2,3), binary tetrahedralCommutator; character tableOpen source

group:binary-tetrahedral#abelian

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Simple

No

GroupNames: SL(2,3), binary tetrahedralChief series; CentreOpen source

group:binary-tetrahedral#simple

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Ordinary complex irreducibles

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

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

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

group:binary-tetrahedral#irreps

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GAP SmallGroups ID

SmallGroup(24,3)

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

group:binary-tetrahedral#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(2,3), binary tetrahedralCharacter tableOpen source

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

group:binary-tetrahedral#representations-2

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Structure and conventions

The center acts trivially on the three singlets and the triplet; those representations descend to A₄.

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

group:binary-tetrahedral#structure-1

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Family

family:binary-polyhedral

GroupNames: SL(2,3), binary tetrahedralHeading; AliasesOpen source

group:binary-tetrahedral#relation-1

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2T / C₂ ≅ A₄

group:a-4

GroupNames: SL(2,3), binary tetrahedralCentre: G/ZOpen source

group:binary-tetrahedral#relation-2

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Binary tetrahedral embedding

group:su-2

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

group:binary-tetrahedral#relation-3

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