{
  "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": "d8",
  "atlasId": "group:d-8",
  "name": "Dihedral group of order eight",
  "symbol": "D₈",
  "latex": "D_8",
  "order": "8",
  "smallGroup": [
    8,
    3
  ],
  "field": "rational",
  "category": "Finite-dimensional ordinary complex representations",
  "coefficientField": "C",
  "characteristic": 0,
  "definition": "D8 = <r,s | r^4 = s^2 = 1, s r s = r^{-1}>; r rotates the square by a quarter turn and s is a reflection.",
  "classConvention": "Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.",
  "derivation": "Four linear characters come from the C2 × C2 abelianization. The geometric doublet has trace 2, -2, 0, 0, 0 in this order. Powers use the displayed presentation.",
  "fieldDefinition": "All character values in this table are rational integers.",
  "classes": [
    {
      "id": "1",
      "label": "1",
      "latex": "1",
      "labelTeX": "1",
      "size": "1",
      "order": 1,
      "representative": "1"
    },
    {
      "id": "2z",
      "label": "r²",
      "latex": "r^2",
      "labelTeX": "r^2",
      "size": "1",
      "order": 2,
      "representative": "r^2"
    },
    {
      "id": "4a",
      "label": "r",
      "latex": "r",
      "labelTeX": "r",
      "size": "2",
      "order": 4,
      "representative": "r"
    },
    {
      "id": "2s",
      "label": "s",
      "latex": "s",
      "labelTeX": "s",
      "size": "2",
      "order": 2,
      "representative": "s"
    },
    {
      "id": "2rs",
      "label": "rs",
      "latex": "rs",
      "labelTeX": "rs",
      "size": "2",
      "order": 2,
      "representative": "rs"
    }
  ],
  "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": "a",
      "label": "a",
      "latex": "a",
      "labelTeX": "a",
      "dimension": "1",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "a",
      "description": "r maps to +1 and s to -1; determinant of the doublet",
      "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": "b",
      "label": "b",
      "latex": "b",
      "labelTeX": "b",
      "dimension": "1",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "b",
      "description": "r maps to -1 and s to +1",
      "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": "c",
      "label": "c",
      "latex": "c",
      "labelTeX": "c",
      "dimension": "1",
      "reality": "Real",
      "indicator": 1,
      "conjugate": "c",
      "description": "r maps to -1 and s to -1",
      "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": "Faithful real-type geometric doublet",
      "characters": [
        {
          "text": "2",
          "latex": "2",
          "coefficients": [
            "2",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "-2",
          "latex": "-2",
          "coefficients": [
            "-2",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        },
        {
          "text": "0",
          "latex": "0",
          "coefficients": [
            "0",
            "0"
          ],
          "basis": "1"
        }
      ]
    }
  ],
  "powerMaps": {
    "1": [
      "1",
      "2z",
      "4a",
      "2s",
      "2rs"
    ],
    "2": [
      "1",
      "1",
      "2z",
      "1",
      "1"
    ],
    "3": [
      "1",
      "2z",
      "4a",
      "2s",
      "2rs"
    ],
    "4": [
      "1",
      "1",
      "1",
      "1",
      "1"
    ],
    "5": [
      "1",
      "2z",
      "4a",
      "2s",
      "2rs"
    ],
    "6": [
      "1",
      "1",
      "2z",
      "1",
      "1"
    ],
    "7": [
      "1",
      "2z",
      "4a",
      "2s",
      "2rs"
    ],
    "8": [
      "1",
      "1",
      "1",
      "1",
      "1"
    ],
    "9": [
      "1",
      "2z",
      "4a",
      "2s",
      "2rs"
    ],
    "10": [
      "1",
      "1",
      "2z",
      "1",
      "1"
    ],
    "11": [
      "1",
      "2z",
      "4a",
      "2s",
      "2rs"
    ],
    "12": [
      "1",
      "1",
      "1",
      "1",
      "1"
    ]
  },
  "sources": [
    {
      "id": "finite-groupnames-d8",
      "title": "Tim Dokchitser, GroupNames: D4",
      "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/D4.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": "d8",
    "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
  }
}