{
  "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/families/finite-ree-g2/",
  "entity": {
    "id": "family:finite-ree-g2",
    "slug": "finite-ree-g2",
    "name": "Small Ree simple groups",
    "symbol": "²G₂(q)",
    "category": "finite",
    "level": "classification",
    "aliases": [
      "Ree(q)"
    ],
    "searchTerms": [
      "Small Ree simple groups",
      "²G₂(q)",
      "Ree(q)"
    ],
    "tags": [
      "Finite simple classification"
    ],
    "summary": "Odd extension degree in characteristic 3; q ≥ 27.",
    "definition": "The Ree twisted groups of type ²G₂.",
    "definitionEvidence": [
      {
        "source": "finite-wilson-suzuki-ree",
        "locator": "Small Ree construction and simplicity; section 5"
      }
    ],
    "properties": [
      {
        "key": "parameters",
        "label": "Parameter domain",
        "value": "q: q = 3^(2m+1), integer m ≥ 1",
        "evidence": [
          {
            "source": "finite-wilson-suzuki-ree",
            "locator": "Small Ree construction and simplicity; section 5"
          }
        ],
        "assumptions": []
      },
      {
        "key": "restriction",
        "label": "Scope and restrictions",
        "value": "Odd extension degree in characteristic 3; q ≥ 27.",
        "evidence": [
          {
            "source": "finite-wilson-suzuki-ree",
            "locator": "Small Ree construction and simplicity; section 5"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "exceptions",
        "title": "Exceptions and identifications",
        "paragraphs": [
          {
            "text": "m = 0; q = 3: ²G₂(3)′ ≅ PSL(2,8), with index 3 in ²G₂(3).",
            "evidence": [
              {
                "source": "finite-craven-identifications",
                "locator": "Section 7, seven exceptional isomorphisms before Proposition 24"
              },
              {
                "source": "finite-wilson-exceptional",
                "locator": "Suzuki–Ree groups: small cases"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [],
    "capabilities": [],
    "kind": "family",
    "data": {
      "parameters": [
        {
          "name": "q",
          "domain": "q = 3^(2m+1), integer m ≥ 1"
        }
      ],
      "exceptions": [
        {
          "at": "m = 0; q = 3",
          "description": "²G₂(3)′ ≅ PSL(2,8), with index 3 in ²G₂(3)."
        }
      ],
      "scope": "Odd extension degree in characteristic 3; q ≥ 27."
    }
  },
  "claims": [
    {
      "id": "family:finite-ree-g2#definition",
      "subject": "family:finite-ree-g2",
      "key": "definition",
      "label": "Definition",
      "value": "The Ree twisted groups of type ²G₂.",
      "evidence": [
        {
          "source": "finite-wilson-suzuki-ree",
          "locator": "Small Ree construction and simplicity; section 5"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "family:finite-ree-g2#parameters",
      "subject": "family:finite-ree-g2",
      "key": "parameters",
      "label": "Parameter domain",
      "value": "q: q = 3^(2m+1), integer m ≥ 1",
      "evidence": [
        {
          "source": "finite-wilson-suzuki-ree",
          "locator": "Small Ree construction and simplicity; section 5"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "family:finite-ree-g2#restriction",
      "subject": "family:finite-ree-g2",
      "key": "restriction",
      "label": "Scope and restrictions",
      "value": "Odd extension degree in characteristic 3; q ≥ 27.",
      "evidence": [
        {
          "source": "finite-wilson-suzuki-ree",
          "locator": "Small Ree construction and simplicity; section 5"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "family:finite-ree-g2#exceptions-1",
      "subject": "family:finite-ree-g2",
      "key": "exceptions-1",
      "label": "Exceptions and identifications",
      "value": "m = 0; q = 3: ²G₂(3)′ ≅ PSL(2,8), with index 3 in ²G₂(3).",
      "evidence": [
        {
          "source": "finite-craven-identifications",
          "locator": "Section 7, seven exceptional isomorphisms before Proposition 24"
        },
        {
          "source": "finite-wilson-exceptional",
          "locator": "Suzuki–Ree groups: small cases"
        }
      ],
      "assumptions": [],
      "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-wilson-suzuki-ree",
      "title": "On the simple groups of Suzuki and Ree",
      "authors": [
        "Robert A. Wilson"
      ],
      "url": "https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/SuzRee0.pdf",
      "kind": "paper",
      "locatorHint": "Construction and simplicity, especially section 5",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-craven-identifications",
      "title": "Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p",
      "authors": [
        "David A. Craven"
      ],
      "url": "https://doi.org/10.1515/jgth-2018-0014",
      "kind": "paper",
      "locatorHint": "Section 7, seven exceptional isomorphisms before Proposition 24",
      "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": "10bbfbdca41312d09252feef8f83a2f6acd617d53fc3661883370e0be8d1fe5e",
    "reviewers": []
  }
}