{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "schema": "2.0.0",
    "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/tits/",
  "entity": {
    "id": "group:tits",
    "slug": "tits",
    "name": "Tits group",
    "symbol": "²F₄(2)′",
    "category": "finite",
    "level": "classification",
    "aliases": [
      "2F4(2)'",
      "Tits",
      "T"
    ],
    "searchTerms": [
      "Tits group",
      "Ree",
      "2F4(2) derived subgroup"
    ],
    "tags": [
      "Finite simple",
      "Lie-type exception",
      "Classification"
    ],
    "summary": "The simple derived subgroup of ²F₄(2), of index two and order 17971200.",
    "definition": "The commutator subgroup ²F₄(2)′ of the smallest large Ree group.",
    "definitionEvidence": [
      {
        "source": "finite-wilson-exceptional",
        "locator": "Suzuki–Ree small cases"
      },
      {
        "source": "finite-atlas-tits",
        "locator": "Heading and standard generators"
      }
    ],
    "properties": [
      {
        "key": "order",
        "label": "Order",
        "value": "17971200",
        "evidence": [
          {
            "source": "finite-atlas-tits",
            "locator": "Order"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Center",
        "value": "1",
        "evidence": [
          {
            "source": "finite-atlas-tits",
            "locator": "Simple group identity"
          }
        ],
        "derivation": {
          "method": "The center of a nonabelian simple group is trivial.",
          "inputs": [
            "group:tits#simple",
            "group:tits#abelian"
          ]
        },
        "assumptions": []
      },
      {
        "key": "simple",
        "label": "Simple",
        "value": "Yes",
        "evidence": [
          {
            "source": "finite-wilson-exceptional",
            "locator": "Suzuki–Ree small cases"
          }
        ],
        "assumptions": []
      },
      {
        "key": "abelian",
        "label": "Abelian",
        "value": "No",
        "evidence": [
          {
            "source": "finite-wilson-exceptional",
            "locator": "Simple Tits group of composite order"
          }
        ],
        "assumptions": []
      },
      {
        "key": "classification-convention",
        "label": "Classification convention",
        "value": "Lie-type small case; excluded from the 26 sporadic groups.",
        "evidence": [
          {
            "source": "finite-atlas-sporadic-index",
            "locator": "Miscellaneous"
          },
          {
            "source": "finite-wilson-exceptional",
            "locator": "Suzuki–Ree small cases"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "scope",
        "title": "Identity and representation scope",
        "paragraphs": [
          {
            "text": "²F₄(2) itself is not simple. Its simple derived subgroup is the Tits group, and the parent group is its extension by an outer involution.",
            "evidence": [
              {
                "source": "finite-atlas-tits",
                "locator": "Standard generators of ²F₄(2) = T.2"
              }
            ],
            "assumptions": []
          },
          {
            "text": "This classification record links to available ATLAS representations. It does not assert a complete local list of ordinary complex irreducibles.",
            "evidence": [
              {
                "source": "finite-atlas-tits",
                "locator": "Representations"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "exceptional-member-of",
        "target": "family:finite-ree-f4",
        "label": "Derived small case of ²F₄",
        "evidence": [
          {
            "source": "finite-wilson-exceptional",
            "locator": "Suzuki–Ree small cases"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "group",
    "data": {
      "category": "finite",
      "order": "17971200",
      "abelian": false,
      "simple": true
    }
  },
  "claims": [
    {
      "id": "group:tits#definition",
      "subject": "group:tits",
      "key": "definition",
      "label": "Definition",
      "value": "The commutator subgroup ²F₄(2)′ of the smallest large Ree group.",
      "evidence": [
        {
          "source": "finite-wilson-exceptional",
          "locator": "Suzuki–Ree small cases"
        },
        {
          "source": "finite-atlas-tits",
          "locator": "Heading and standard generators"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#order",
      "subject": "group:tits",
      "key": "order",
      "label": "Order",
      "value": "17971200",
      "evidence": [
        {
          "source": "finite-atlas-tits",
          "locator": "Order"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#center",
      "subject": "group:tits",
      "key": "center",
      "label": "Center",
      "value": "1",
      "evidence": [
        {
          "source": "finite-atlas-tits",
          "locator": "Simple group identity"
        }
      ],
      "derivation": {
        "method": "The center of a nonabelian simple group is trivial.",
        "inputs": [
          "group:tits#simple",
          "group:tits#abelian"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#simple",
      "subject": "group:tits",
      "key": "simple",
      "label": "Simple",
      "value": "Yes",
      "evidence": [
        {
          "source": "finite-wilson-exceptional",
          "locator": "Suzuki–Ree small cases"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#abelian",
      "subject": "group:tits",
      "key": "abelian",
      "label": "Abelian",
      "value": "No",
      "evidence": [
        {
          "source": "finite-wilson-exceptional",
          "locator": "Simple Tits group of composite order"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#classification-convention",
      "subject": "group:tits",
      "key": "classification-convention",
      "label": "Classification convention",
      "value": "Lie-type small case; excluded from the 26 sporadic groups.",
      "evidence": [
        {
          "source": "finite-atlas-sporadic-index",
          "locator": "Miscellaneous"
        },
        {
          "source": "finite-wilson-exceptional",
          "locator": "Suzuki–Ree small cases"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#scope-1",
      "subject": "group:tits",
      "key": "scope-1",
      "label": "Identity and representation scope",
      "value": "²F₄(2) itself is not simple. Its simple derived subgroup is the Tits group, and the parent group is its extension by an outer involution.",
      "evidence": [
        {
          "source": "finite-atlas-tits",
          "locator": "Standard generators of ²F₄(2) = T.2"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#scope-2",
      "subject": "group:tits",
      "key": "scope-2",
      "label": "Identity and representation scope",
      "value": "This classification record links to available ATLAS representations. It does not assert a complete local list of ordinary complex irreducibles.",
      "evidence": [
        {
          "source": "finite-atlas-tits",
          "locator": "Representations"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:tits#relation-1",
      "subject": "group:tits",
      "key": "relation-1",
      "label": "Derived small case of ²F₄",
      "value": "family:finite-ree-f4",
      "evidence": [
        {
          "source": "finite-wilson-exceptional",
          "locator": "Suzuki–Ree small cases"
        }
      ],
      "assumptions": [],
      "relationship": "exceptional-member-of",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "finite-wilson-exceptional",
      "title": "Finite simple groups: Lecture 4 — Exceptional groups",
      "authors": [
        "Robert A. Wilson"
      ],
      "url": "https://webspace.maths.qmul.ac.uk/r.a.wilson/FSG/lect4.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Ten exceptional families; Suzuki–Ree parameters and small cases; G2",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-atlas-sporadic-index",
      "title": "ATLAS of Finite Group Representations: Sporadic groups",
      "authors": [
        "Robert A. Wilson",
        "ATLAS of Finite Group Representations contributors"
      ],
      "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/",
      "kind": "database",
      "locatorHint": "Sporadic group index; Tits group listed separately under Miscellaneous",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-atlas-tits",
      "title": "ATLAS: Tits group ²F₄(2)′",
      "authors": [
        "ATLAS of Finite Group Representations contributors"
      ],
      "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/",
      "kind": "database",
      "locatorHint": "Order; standard generators; ²F₄(2) extension",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "724541a75e63b26a314119c6c5b534acc13a00717de484f938423f2b533fcc1c",
    "reviewers": []
  }
}