{
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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-octahedral/",
  "entity": {
    "id": "group:binary-octahedral",
    "slug": "binary-octahedral",
    "name": "Binary octahedral group",
    "symbol": "2O",
    "category": "finite",
    "level": "reference",
    "aliases": [
      "CSU(2,3)",
      "Binary octahedral",
      "2O"
    ],
    "searchTerms": [
      "Binary octahedral group",
      "2O",
      "CSU(2,3)",
      "Binary octahedral",
      "2O",
      "SmallGroup(48,28)"
    ],
    "tags": [
      "Finite reference",
      "Nonabelian"
    ],
    "summary": "The binary rotation group of the cube or octahedron, a central double cover of S₄.",
    "definition": "The inverse image of the octahedral rotation group in SU(2).",
    "definitionEvidence": [
      {
        "source": "finite-taylor-mckay",
        "locator": "Binary polyhedral groups"
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      {
        "source": "finite-groupnames-binary-octahedral",
        "locator": "Aliases"
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    ],
    "properties": [
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        "key": "order",
        "label": "Order",
        "value": "48",
        "evidence": [
          {
            "source": "finite-groupnames-binary-octahedral",
            "locator": "Heading: order"
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        ],
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      },
      {
        "key": "center",
        "label": "Center",
        "value": "C₂",
        "evidence": [
          {
            "source": "finite-groupnames-binary-octahedral",
            "locator": "Centre"
          }
        ],
        "assumptions": []
      },
      {
        "key": "abelian",
        "label": "Abelian",
        "value": "No",
        "evidence": [
          {
            "source": "finite-groupnames-binary-octahedral",
            "locator": "Commutator; character table"
          }
        ],
        "assumptions": []
      },
      {
        "key": "simple",
        "label": "Simple",
        "value": "No",
        "evidence": [
          {
            "source": "finite-groupnames-binary-octahedral",
            "locator": "Chief series; Centre"
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        ],
        "assumptions": []
      },
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        "key": "irreps",
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        "value": "1, 1, 2, 2, 2, 3, 3, 4",
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            "source": "finite-groupnames-binary-octahedral",
            "locator": "Character table: identity column and real/complex/symplectic annotations"
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            "source": "gap",
            "locator": "Irr and section 71.12-5 Indicator"
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        ],
        "assumptions": [
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          "Dimensions listed with multiplicity"
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      },
      {
        "key": "smallgroups-id",
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        "evidence": [
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            "source": "finite-groupnames-binary-octahedral",
            "locator": "Aliases; GAP/Magma code by ID"
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        ],
        "assumptions": []
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    ],
    "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": [
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                "source": "finite-groupnames-binary-octahedral",
                "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-octahedral",
                "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 faithful doublets and the four-dimensional irrep are quaternionic. The other irreps descend to the rotational quotient S₄.",
            "evidence": [
              {
                "source": "finite-groupnames-binary-octahedral",
                "locator": "Heading; Centre; group-property labels; Aliases"
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            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
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        "type": "member-of",
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        "label": "Family",
        "evidence": [
          {
            "source": "finite-groupnames-binary-octahedral",
            "locator": "Heading; Aliases"
          }
        ]
      },
      {
        "type": "central-quotient",
        "target": "group:s-4",
        "label": "2O / C₂ ≅ S₄",
        "evidence": [
          {
            "source": "finite-groupnames-binary-octahedral",
            "locator": "Centre: G/Z"
          }
        ]
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      {
        "type": "subgroup-of",
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        "label": "Binary octahedral embedding",
        "evidence": [
          {
            "source": "finite-taylor-mckay",
            "locator": "Binary polyhedral groups"
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      }
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    "capabilities": [],
    "kind": "group",
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      "order": "48",
      "abelian": false,
      "simple": false,
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        "category": "ordinary-complex",
        "complete": true,
        "labels": [
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          "ρ2",
          "ρ3",
          "ρ4",
          "ρ5",
          "ρ6",
          "ρ7",
          "ρ8"
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        "dimensions": [
          1,
          1,
          2,
          2,
          2,
          3,
          3,
          4
        ],
        "reality": [
          "Real",
          "Real",
          "Real",
          "Quaternionic",
          "Quaternionic",
          "Real",
          "Real",
          "Quaternionic"
        ]
      }
    }
  },
  "claims": [
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      "id": "group:binary-octahedral#definition",
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      "label": "Definition",
      "value": "The inverse image of the octahedral rotation group in SU(2).",
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        {
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          "locator": "Binary polyhedral groups"
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      "assumptions": [],
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      "revision": "2026.11-workbench.4"
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      "id": "group:binary-octahedral#order",
      "subject": "group:binary-octahedral",
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      "id": "group:binary-octahedral#center",
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      "label": "Center",
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      "id": "group:binary-octahedral#abelian",
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      "revision": "2026.11-workbench.4"
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      "id": "group:binary-octahedral#simple",
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          "source": "finite-groupnames-binary-octahedral",
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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          "source": "finite-groupnames-binary-octahedral",
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          "locator": "Irr and section 71.12-5 Indicator"
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      ],
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
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      "label": "GAP SmallGroups ID",
      "value": "SmallGroup(48,28)",
      "evidence": [
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          "source": "finite-groupnames-binary-octahedral",
          "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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      "id": "group:binary-octahedral#representations-1",
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      "key": "representations-2",
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        {
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      "revision": "2026.11-workbench.4"
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      "reviewStatus": "pending-expert-review",
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    },
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      "id": "finite-taylor-mckay",
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      "id": "finite-groupnames-binary-octahedral",
      "title": "GroupNames: CSU(2,3), binary octahedral",
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