{
  "schema": "groups.qft.org/entity/2",
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    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
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  "canonicalURL": "https://groups.qft.org/atlas/groups/q-8/",
  "entity": {
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    "slug": "q-8",
    "name": "Quaternion group",
    "symbol": "Q₈",
    "category": "finite",
    "level": "executable",
    "aliases": [
      "Quaternion group of order 8"
    ],
    "searchTerms": [],
    "tags": [
      "Spin cover",
      "McKay"
    ],
    "summary": "The eight unit quaternions {±1, ±i, ±j, ±k}, naturally a finite subgroup of SU(2).",
    "definition": "i² = j² = k² = ijk = −1",
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        "key": "center",
        "label": "Center",
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            "locator": "Centre"
          }
        ],
        "assumptions": []
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      {
        "key": "abelian",
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            "source": "gap",
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          }
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        "assumptions": []
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        "id": "physics",
        "title": "In physics",
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          {
            "text": "Projective symmetries, quaternionic representations, and the affine D₄ McKay graph.",
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                "source": "mckay",
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        "label": "Related group"
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        "type": "related",
        "target": "group:d-8",
        "label": "Related group"
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      {
        "type": "member-of",
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        "dimensions": [
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          1,
          1,
          1,
          2
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        "reality": [
          "Real",
          "Real",
          "Real",
          "Real",
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      },
      "capabilityPresets": {
        "finite-characters": {
          "group": "q8",
          "left": "2",
          "right": "2"
        }
      }
    }
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      "id": "groupnames-q-8",
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        "Tim Dokchitser"
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    "currentReviews": 0,
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