FiniteGroupData
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ResourceFunction["FindGroupIsomorphism"]
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isomorphicGroupsQ[group1_,group2_]:=Length[ResourceFunction["FindGroupIsomorphism"][group1,group2,1]]1
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FindGroup[perms_]:=FindGroup[perms]=Select[FiniteGroupData[GroupOrder[perms]],isomorphicGroupsQ[perms,FiniteGroupData[#,"PermutationGroupRepresentation"]]&]
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Values[{FiniteGroupData[First[FindGroup[#[[1,-1]]]],"Notation"],GroupOrder[#[[1,-1]]],Length[#],WolframModelPlot[First[#],ImageSize45]&/@Take[#,UpTo[2]]}&/@GroupBy[#HypergraphAutomorphismGroup[#]&/@EnumerateHypergraphs[{{3,2}}],Last]]
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groupsgrid[sig_]:=Labeled[Grid[Values[{FiniteGroupData[#[[1,-1]],"Notation"],FiniteGroupData[#[[1,-1]],"Order"],Length[#],Row[WolframModelPlot[First[#],ImageSize30]&/@Take[#,UpTo[3]],Spacer[2]]}&/@GroupBy[#First[FindGroup[HypergraphAutomorphismGroup[#]]]&/@EnumerateHypergraphs[sig],Last]],FrameAll,Alignment{{Center,Right,Right,Center}}],Row[Subscript@@@sig]]
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groupsgrid[{{3,2}}]
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FiniteGroupData[{"SymmetricGroup",3},"Notation"]
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S
3