{
  "schema": "groups.qft.org/character-table/1",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "id": "s4",
  "atlasId": "group:s-4",
  "name": "Symmetric group on four letters",
  "symbol": "S₄",
  "latex": "S_4",
  "order": "24",
  "smallGroup": [
    24,
    12
  ],
  "field": "rational",
  "category": "Finite-dimensional ordinary complex representations",
  "coefficientField": "C",
  "characteristic": 0,
  "definition": "Permutations of four objects.",
  "classConvention": "Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.",
  "derivation": "The doublet factors through the action on the three pairings of four letters. The standard triplet has character equal to the number of fixed letters minus one; its sign twist gives the other triplet.",
  "fieldDefinition": "All character values in this table are rational integers.",
  "classes": [
    {
      "id": "1",
      "label": "1",
      "latex": "1",
      "labelTeX": "1",
      "size": "1",
      "order": 1,
      "representative": "()"
    },
    {
      "id": "2a",
      "label": "(12)",
      "latex": "(12)",
      "labelTeX": "(12)",
      "size": "6",
      "order": 2,
      "representative": "(12)"
    },
    {
      "id": "2b",
      "label": "(12)(34)",
      "latex": "(12)(34)",
      "labelTeX": "(12)(34)",
      "size": "3",
      "order": 2,
      "representative": "(12)(34)"
    },
    {
      "id": "3a",
      "label": "(123)",
      "latex": "(123)",
      "labelTeX": "(123)",
      "size": "8",
      "order": 3,
      "representative": "(123)"
    },
    {
      "id": "4a",
      "label": "(1234)",
      "latex": "(1234)",
      "labelTeX": "(1234)",
      "size": "6",
      "order": 4,
      "representative": "(1234)"
    }
  ],
  "irreps": [
    {
      "id": "1",
      "label": "1",
      "latex": "\\mathbf{1}",
      "labelTeX": "\\mathbf{1}",
      "dimension": "1",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "1",
      "description": "Trivial representation",
      "characters": [
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        }
      ]
    },
    {
      "id": "sgn",
      "label": "sgn",
      "latex": "\\mathrm{sgn}",
      "labelTeX": "\\mathrm{sgn}",
      "dimension": "1",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "sgn",
      "description": "Sign of a permutation",
      "characters": [
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        }
      ]
    },
    {
      "id": "2",
      "label": "2",
      "latex": "\\mathbf{2}",
      "labelTeX": "\\mathbf{2}",
      "dimension": "2",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "2",
      "description": "Standard S3 doublet pulled back through S4 → S3",
      "characters": [
        {
          "text": "2",
          "latex": "2",
          "coefficients": [
            "2",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "2",
          "latex": "2",
          "coefficients": [
            "2",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        }
      ]
    },
    {
      "id": "3",
      "label": "3",
      "latex": "\\mathbf{3}",
      "labelTeX": "\\mathbf{3}",
      "dimension": "3",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "3",
      "description": "Standard triplet; the determinant is sgn",
      "characters": [
        {
          "text": "3",
          "latex": "3",
          "coefficients": [
            "3",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        }
      ]
    },
    {
      "id": "3p",
      "label": "3′",
      "latex": "\\mathbf{3}'",
      "labelTeX": "\\mathbf{3}'",
      "dimension": "3",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "3p",
      "description": "Sign twist of the standard triplet; the cubic-rotation representation",
      "characters": [
        {
          "text": "3",
          "latex": "3",
          "coefficients": [
            "3",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-1",
          "latex": "-1",
          "coefficients": [
            "-1",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "1",
          "latex": "1",
          "coefficients": [
            "1",
            "0"
          ],
          "basis": "1"
        }
      ]
    }
  ],
  "powerMaps": {
    "1": [
      "1",
      "2a",
      "2b",
      "3a",
      "4a"
    ],
    "2": [
      "1",
      "1",
      "1",
      "3a",
      "2b"
    ],
    "3": [
      "1",
      "2a",
      "2b",
      "1",
      "4a"
    ],
    "4": [
      "1",
      "1",
      "1",
      "3a",
      "1"
    ],
    "5": [
      "1",
      "2a",
      "2b",
      "3a",
      "4a"
    ],
    "6": [
      "1",
      "1",
      "1",
      "1",
      "2b"
    ],
    "7": [
      "1",
      "2a",
      "2b",
      "3a",
      "4a"
    ],
    "8": [
      "1",
      "1",
      "1",
      "3a",
      "1"
    ],
    "9": [
      "1",
      "2a",
      "2b",
      "1",
      "4a"
    ],
    "10": [
      "1",
      "1",
      "1",
      "3a",
      "2b"
    ],
    "11": [
      "1",
      "2a",
      "2b",
      "3a",
      "4a"
    ],
    "12": [
      "1",
      "1",
      "1",
      "1",
      "1"
    ]
  },
  "sources": [
    {
      "id": "finite-groupnames-s4",
      "title": "Tim Dokchitser, GroupNames: S4",
      "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/S4.html",
      "locator": "Character table, class sizes, group presentation, and representation type. Our explicit representatives and row labels fix any reordering.",
      "kind": "database",
      "checked": "2026-11-08",
      "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content."
    },
    {
      "id": "finite-etingof",
      "title": "Etingof et al., Introduction to Representation Theory",
      "url": "https://math.mit.edu/~etingof/reprbook.pdf",
      "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators",
      "kind": "lecture-notes",
      "checked": "2026-11-08",
      "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content."
    },
    {
      "id": "finite-gap-characters",
      "title": "GAP Reference Manual, Chapter 72: Class Functions",
      "url": "https://gap-system.github.io/gap/doc/ref/chap72_mj.html",
      "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers",
      "kind": "documentation",
      "checked": "2026-11-08",
      "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content."
    },
    {
      "id": "finite-gap-tables",
      "title": "GAP Reference Manual, Chapter 71: Character Tables",
      "url": "https://gap-system.github.io/gap/doc/ref/chap71_mj.html",
      "locator": "Section 71.12-5: second Frobenius–Schur indicator; class data and power maps",
      "kind": "documentation",
      "checked": "2026-11-08",
      "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content."
    },
    {
      "id": "finite-gap-power-maps",
      "title": "GAP Reference Manual, Chapter 73: Maps Concerning Character Tables",
      "url": "https://gap-system.github.io/gap/doc/ref/chap73_mj.html",
      "locator": "Section 73.1: power maps are extra class data and need not follow from the character matrix alone",
      "kind": "documentation",
      "checked": "2026-11-08",
      "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content."
    }
  ],
  "conventionId": "finite-ordinary-complex-v1",
  "conventionVersion": "1.0.0",
  "convention": {
    "id": "finite-ordinary-complex-v1",
    "version": "1.0.0",
    "field": "Complex numbers; characteristic zero",
    "category": "Finite-dimensional ordinary complex representations",
    "innerProduct": "sum_C |C| chi(C) conjugate(psi(C)) / |G|",
    "dihedralNotation": "D8 is the symmetry group of the square, of order eight; GroupNames calls it D4.",
    "omega": "exp(2 pi i / 3)",
    "phi": "(1 + sqrt(5)) / 2",
    "representationLabels": "Labels and conjugacy-class orders are local conventions, recorded with each table. They must not be identified with upstream row numbers.",
    "scope": "Character decompositions do not specify Clebsch–Gordan coefficients, representation matrices, or a complete Lorentz-invariant operator basis."
  },
  "algorithmVersion": "finite-characters-1.0.0",
  "dataVersion": "small-characters-1.0.0",
  "checks": {
    "group": "s4",
    "rows": 5,
    "classes": 5,
    "classSizes": true,
    "dimensionSquares": true,
    "rowOrthogonality": true,
    "columnOrthogonality": true,
    "frobeniusSchur": true,
    "conjugates": true,
    "powerMaps": true
  },
  "independentExpertReview": "pending",
  "limits": {
    "maxDegree": 12,
    "maxInputMultiplicity": 12,
    "maxInputDimension": 60
  }
}