{
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    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
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  "canonicalURL": "https://groups.qft.org/atlas/groups/s-3/",
  "entity": {
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    "slug": "s-3",
    "name": "Symmetric group on three letters",
    "symbol": "S₃",
    "category": "finite",
    "level": "executable",
    "aliases": [
      "D3",
      "D6",
      "Triangle symmetries"
    ],
    "searchTerms": [],
    "tags": [
      "Permutations",
      "Discrete symmetry"
    ],
    "summary": "The permutation group of three objects, also the full symmetry group of an equilateral triangle.",
    "definition": "⟨r,s | r³ = s² = 1, srs = r⁻¹⟩",
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        "label": "Center",
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            "locator": "Centre"
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        "assumptions": []
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        "label": "Abelian",
        "value": "No",
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            "source": "groupnames-s-3",
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        "assumptions": []
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        "key": "simple",
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          }
        ],
        "assumptions": []
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        "title": "Representations",
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            "text": "There are three ordinary complex irreps: the trivial and sign representations, and a real-type doublet.",
            "evidence": [
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            "assumptions": []
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      {
        "id": "physics",
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        "paragraphs": [
          {
            "text": "Permutation symmetry and a compact first example of nonabelian character theory.",
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                "locator": "Character table; matrix and permutation representations"
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        "label": "Related group"
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        "type": "related",
        "target": "group:a-4",
        "label": "Related group"
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      {
        "type": "member-of",
        "target": "family:finite-symmetric",
        "label": "Symmetric family",
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        "type": "member-of",
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      "order": "6",
      "abelian": false,
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        "complete": true,
        "labels": [
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        "dimensions": [
          1,
          1,
          2
        ],
        "reality": [
          "Real",
          "Real",
          "Real"
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      "capabilityPresets": {
        "finite-characters": {
          "group": "s3",
          "left": "2",
          "right": "2"
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