{
  "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/delta-27/",
  "entity": {
    "id": "group:delta-27",
    "slug": "delta-27",
    "name": "Delta(27), the finite Heisenberg group",
    "symbol": "Δ(27)",
    "category": "finite",
    "level": "reference",
    "aliases": [
      "Delta(27)",
      "He3",
      "Heisenberg group of order 27",
      "3_+^(1+2)",
      "(C3 × C3) ⋊ C3"
    ],
    "searchTerms": [
      "Delta(27), the finite Heisenberg group",
      "Δ(27)",
      "Delta(27)",
      "He3",
      "Heisenberg group of order 27",
      "3_+^(1+2)",
      "(C3 × C3) ⋊ C3",
      "SmallGroup(27,3)"
    ],
    "tags": [
      "Finite reference",
      "Nonabelian"
    ],
    "summary": "The nonabelian Heisenberg group over F₃, with nine singlets and two conjugate triplets.",
    "definition": "The order-27 subgroup of SU(3) generated by diag(ω,1,ω⁻¹), diag(1,ω,ω⁻¹) and the cyclic coordinate permutation, with ω³ = 1 and ω ≠ 1.",
    "definitionEvidence": [
      {
        "source": "finite-benvenuti-delta27",
        "locator": "Equation (3.12)"
      },
      {
        "source": "finite-groupnames-delta-27",
        "locator": "Heading and Aliases"
      }
    ],
    "properties": [
      {
        "key": "order",
        "label": "Order",
        "value": "27",
        "evidence": [
          {
            "source": "finite-groupnames-delta-27",
            "locator": "Heading: order"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Center",
        "value": "C₃",
        "evidence": [
          {
            "source": "finite-groupnames-delta-27",
            "locator": "Centre"
          }
        ],
        "assumptions": []
      },
      {
        "key": "abelian",
        "label": "Abelian",
        "value": "No",
        "evidence": [
          {
            "source": "finite-groupnames-delta-27",
            "locator": "Commutator; character table"
          }
        ],
        "assumptions": []
      },
      {
        "key": "simple",
        "label": "Simple",
        "value": "No",
        "evidence": [
          {
            "source": "finite-groupnames-delta-27",
            "locator": "Chief series; Centre"
          }
        ],
        "assumptions": []
      },
      {
        "key": "irreps",
        "label": "Ordinary complex irreducibles",
        "value": "1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3",
        "evidence": [
          {
            "source": "finite-groupnames-delta-27",
            "locator": "Character table: identity column and real/complex/symplectic annotations"
          },
          {
            "source": "gap",
            "locator": "Irr and section 71.12-5 Indicator"
          }
        ],
        "assumptions": [
          "Characteristic zero",
          "Finite-dimensional irreducibles over C",
          "Dimensions listed with multiplicity"
        ]
      },
      {
        "key": "smallgroups-id",
        "label": "GAP SmallGroups ID",
        "value": "SmallGroup(27,3)",
        "evidence": [
          {
            "source": "finite-groupnames-delta-27",
            "locator": "Aliases; GAP/Magma code by ID"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
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        "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": [
              {
                "source": "finite-groupnames-delta-27",
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            "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-delta-27",
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              },
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                "locator": "Irr and section 71.12-5 Indicator"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "structure",
        "title": "Structure and conventions",
        "paragraphs": [
          {
            "text": "The center and commutator subgroup are both C₃; the abelianization is C₃ × C₃.",
            "evidence": [
              {
                "source": "finite-groupnames-delta-27",
                "locator": "Centre; Commutator"
              }
            ],
            "assumptions": []
          },
          {
            "text": "The SU(3) matrix realization is used in the cited study of invariant rings and orbifold gauge theories.",
            "evidence": [
              {
                "source": "finite-benvenuti-delta27",
                "locator": "Section 3, equation (3.12) and following discussion"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "subgroup-of",
        "target": "group:su-3",
        "label": "Specified SU(3) embedding",
        "evidence": [
          {
            "source": "finite-benvenuti-delta27",
            "locator": "Equation (3.12)"
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        ]
      }
    ],
    "capabilities": [],
    "kind": "group",
    "data": {
      "category": "finite",
      "order": "27",
      "abelian": false,
      "simple": false,
      "irreps": {
        "category": "ordinary-complex",
        "complete": true,
        "labels": [
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          "ρ2",
          "ρ3",
          "ρ4",
          "ρ5",
          "ρ6",
          "ρ7",
          "ρ8",
          "ρ9",
          "ρ10",
          "ρ11"
        ],
        "dimensions": [
          1,
          1,
          1,
          1,
          1,
          1,
          1,
          1,
          1,
          3,
          3
        ],
        "reality": [
          "Real",
          "Complex",
          "Complex",
          "Complex",
          "Complex",
          "Complex",
          "Complex",
          "Complex",
          "Complex",
          "Complex",
          "Complex"
        ]
      }
    }
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  "claims": [
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      "id": "group:delta-27#definition",
      "subject": "group:delta-27",
      "key": "definition",
      "label": "Definition",
      "value": "The order-27 subgroup of SU(3) generated by diag(ω,1,ω⁻¹), diag(1,ω,ω⁻¹) and the cyclic coordinate permutation, with ω³ = 1 and ω ≠ 1.",
      "evidence": [
        {
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          "locator": "Equation (3.12)"
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        }
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:delta-27#center",
      "subject": "group:delta-27",
      "key": "center",
      "label": "Center",
      "value": "C₃",
      "evidence": [
        {
          "source": "finite-groupnames-delta-27",
          "locator": "Centre"
        }
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:delta-27#abelian",
      "subject": "group:delta-27",
      "key": "abelian",
      "label": "Abelian",
      "value": "No",
      "evidence": [
        {
          "source": "finite-groupnames-delta-27",
          "locator": "Commutator; character table"
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "id": "group:delta-27#simple",
      "subject": "group:delta-27",
      "key": "simple",
      "label": "Simple",
      "value": "No",
      "evidence": [
        {
          "source": "finite-groupnames-delta-27",
          "locator": "Chief series; Centre"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:delta-27#irreps",
      "subject": "group:delta-27",
      "key": "irreps",
      "label": "Ordinary complex irreducibles",
      "value": "1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3",
      "evidence": [
        {
          "source": "finite-groupnames-delta-27",
          "locator": "Character table: identity column and real/complex/symplectic annotations"
        },
        {
          "source": "gap",
          "locator": "Irr and section 71.12-5 Indicator"
        }
      ],
      "assumptions": [
        "Characteristic zero",
        "Finite-dimensional irreducibles over C",
        "Dimensions listed with multiplicity"
      ],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:delta-27#smallgroups-id",
      "subject": "group:delta-27",
      "key": "smallgroups-id",
      "label": "GAP SmallGroups ID",
      "value": "SmallGroup(27,3)",
      "evidence": [
        {
          "source": "finite-groupnames-delta-27",
          "locator": "Aliases; GAP/Magma code by ID"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:delta-27#representations-1",
      "subject": "group:delta-27",
      "key": "representations-1",
      "label": "Representation scope",
      "value": "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": [
        {
          "source": "finite-groupnames-delta-27",
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      "revision": "2026.11-workbench.4"
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      "id": "group:delta-27#representations-2",
      "subject": "group:delta-27",
      "key": "representations-2",
      "label": "Representation scope",
      "value": "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-delta-27",
          "locator": "Character table: identity column and real/complex/symplectic annotations"
        },
        {
          "source": "gap",
          "locator": "Irr and section 71.12-5 Indicator"
        }
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:delta-27#structure-1",
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "label": "Specified SU(3) embedding",
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          "source": "finite-benvenuti-delta27",
          "locator": "Equation (3.12)"
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      "relationship": "subgroup-of",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "id": "gap",
      "title": "GAP · Characters and symmetrizations",
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      "url": "https://gap-system.github.io/gap/doc/ref/chap72_mj.html",
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      "locatorHint": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "finite-benvenuti-delta27",
      "title": "Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics",
      "authors": [
        "Sergio Benvenuti",
        "Bo Feng",
        "Amihay Hanany",
        "Yang-Hui He"
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      "url": "https://arxiv.org/abs/hep-th/0608050",
      "kind": "paper",
      "locatorHint": "Section 3, equation (3.12): Delta(27) matrix generators and Heisenberg identification",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-groupnames-delta-27",
      "title": "GroupNames: He₃",
      "authors": [
        "Tim Dokchitser"
      ],
      "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/He3.html",
      "kind": "database",
      "locatorHint": "Order, Centre, Aliases, generators and relations, Character table",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    }
  ],
  "review": {
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    "label": "Expert review pending",
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