{
  "schema": "groups.qft.org/entity/2",
  "versions": {
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    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/groups/binary-tetrahedral/",
  "entity": {
    "id": "group:binary-tetrahedral",
    "slug": "binary-tetrahedral",
    "name": "Binary tetrahedral group",
    "symbol": "2T",
    "category": "finite",
    "level": "reference",
    "aliases": [
      "SL(2,3)",
      "2.A4",
      "Q8 ⋊ C3"
    ],
    "searchTerms": [
      "Binary tetrahedral group",
      "2T",
      "SL(2,3)",
      "2.A4",
      "Q8 ⋊ C3",
      "SmallGroup(24,3)"
    ],
    "tags": [
      "Finite reference",
      "Nonabelian"
    ],
    "summary": "The central double cover of A₄, with three doublets of different Frobenius–Schur types.",
    "definition": "The inverse image of the tetrahedral rotation group in SU(2); abstractly SL(2,3).",
    "definitionEvidence": [
      {
        "source": "finite-taylor-mckay",
        "locator": "Binary polyhedral groups; binary tetrahedral example"
      },
      {
        "source": "finite-groupnames-binary-tetrahedral",
        "locator": "Aliases"
      }
    ],
    "properties": [
      {
        "key": "order",
        "label": "Order",
        "value": "24",
        "evidence": [
          {
            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Heading: order"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Center",
        "value": "C₂",
        "evidence": [
          {
            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Centre"
          }
        ],
        "assumptions": []
      },
      {
        "key": "abelian",
        "label": "Abelian",
        "value": "No",
        "evidence": [
          {
            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Commutator; character table"
          }
        ],
        "assumptions": []
      },
      {
        "key": "simple",
        "label": "Simple",
        "value": "No",
        "evidence": [
          {
            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Chief series; Centre"
          }
        ],
        "assumptions": []
      },
      {
        "key": "irreps",
        "label": "Ordinary complex irreducibles",
        "value": "1, 1, 1, 2, 2, 2, 3",
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            "source": "finite-groupnames-binary-tetrahedral",
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          },
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            "source": "gap",
            "locator": "Irr and section 71.12-5 Indicator"
          }
        ],
        "assumptions": [
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          "Finite-dimensional irreducibles over C",
          "Dimensions listed with multiplicity"
        ]
      },
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        "key": "smallgroups-id",
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        "evidence": [
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            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Aliases; GAP/Magma code by ID"
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        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "representations",
        "title": "Representation scope",
        "paragraphs": [
          {
            "text": "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.",
            "evidence": [
              {
                "source": "finite-groupnames-binary-tetrahedral",
                "locator": "Character table"
              }
            ],
            "assumptions": []
          },
          {
            "text": "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.",
            "evidence": [
              {
                "source": "finite-groupnames-binary-tetrahedral",
                "locator": "Character table: identity column and real/complex/symplectic annotations"
              },
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                "locator": "Irr and section 71.12-5 Indicator"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "structure",
        "title": "Structure and conventions",
        "paragraphs": [
          {
            "text": "The center acts trivially on the three singlets and the triplet; those representations descend to A₄.",
            "evidence": [
              {
                "source": "finite-groupnames-binary-tetrahedral",
                "locator": "Heading; Centre; group-property labels; Aliases"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "member-of",
        "target": "family:binary-polyhedral",
        "label": "Family",
        "evidence": [
          {
            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Heading; Aliases"
          }
        ]
      },
      {
        "type": "central-quotient",
        "target": "group:a-4",
        "label": "2T / C₂ ≅ A₄",
        "evidence": [
          {
            "source": "finite-groupnames-binary-tetrahedral",
            "locator": "Centre: G/Z"
          }
        ]
      },
      {
        "type": "subgroup-of",
        "target": "group:su-2",
        "label": "Binary tetrahedral embedding",
        "evidence": [
          {
            "source": "finite-taylor-mckay",
            "locator": "Binary polyhedral groups"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "group",
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      "order": "24",
      "abelian": false,
      "simple": false,
      "irreps": {
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        "complete": true,
        "labels": [
          "ρ1",
          "ρ2",
          "ρ3",
          "ρ4",
          "ρ5",
          "ρ6",
          "ρ7"
        ],
        "dimensions": [
          1,
          1,
          1,
          2,
          2,
          2,
          3
        ],
        "reality": [
          "Real",
          "Complex",
          "Complex",
          "Quaternionic",
          "Complex",
          "Complex",
          "Real"
        ]
      }
    }
  },
  "claims": [
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      "id": "group:binary-tetrahedral#definition",
      "subject": "group:binary-tetrahedral",
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      "label": "Definition",
      "value": "The inverse image of the tetrahedral rotation group in SU(2); abstractly SL(2,3).",
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      "reviewStatus": "pending-expert-review",
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      "id": "group:binary-tetrahedral#order",
      "subject": "group:binary-tetrahedral",
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      "id": "group:binary-tetrahedral#center",
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      "assumptions": [],
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      "id": "group:binary-tetrahedral#abelian",
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      "reviewStatus": "pending-expert-review",
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          "locator": "Chief series; Centre"
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "id": "group:binary-tetrahedral#irreps",
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        {
          "source": "finite-groupnames-binary-tetrahedral",
          "locator": "Character table: identity column and real/complex/symplectic annotations"
        },
        {
          "source": "gap",
          "locator": "Irr and section 71.12-5 Indicator"
        }
      ],
      "assumptions": [
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        "Dimensions listed with multiplicity"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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    {
      "id": "group:binary-tetrahedral#smallgroups-id",
      "subject": "group:binary-tetrahedral",
      "key": "smallgroups-id",
      "label": "GAP SmallGroups ID",
      "value": "SmallGroup(24,3)",
      "evidence": [
        {
          "source": "finite-groupnames-binary-tetrahedral",
          "locator": "Aliases; GAP/Magma code by ID"
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "subject": "group:binary-tetrahedral",
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      "label": "Representation scope",
      "value": "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.",
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      "revision": "2026.11-workbench.4"
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    {
      "id": "group:binary-tetrahedral#representations-2",
      "subject": "group:binary-tetrahedral",
      "key": "representations-2",
      "label": "Representation scope",
      "value": "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.",
      "evidence": [
        {
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      "revision": "2026.11-workbench.4"
    },
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      "id": "group:binary-tetrahedral#structure-1",
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      "key": "structure-1",
      "label": "Structure and conventions",
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        {
          "source": "finite-groupnames-binary-tetrahedral",
          "locator": "Heading; Centre; group-property labels; Aliases"
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "revision": "2026.11-workbench.4"
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      "revision": "2026.11-workbench.4"
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      "locatorHint": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "finite-taylor-mckay",
      "title": "Character Theory of Finite Groups: Day 4 — The McKay Correspondence",
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      "kind": "lecture-notes",
      "locatorHint": "Binary polyhedral groups and the SU(2) → SO(3) cover, slides 7–8",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-groupnames-binary-tetrahedral",
      "title": "GroupNames: SL(2,3), binary tetrahedral",
      "authors": [
        "Tim Dokchitser"
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      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    }
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    "label": "Expert review pending",
    "currentReviews": 0,
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