Introduction

 
Database of Simple Groups
      SimpleGroup(N) : RngIntElt -> Grp
      SimpleGroupOfOrder(M) : RngIntElt -> Grp
      NumberOfSimpleGroups() : -> RngIntElt
      SimpleGroupName(N) : RngIntElt -> MonStgElt
      SimpleGroupNameToNumber(S) : MonStgElt -> RngIntElt
      SimpleGroupIDToNumber(T) : Tup -> RngIntElt
      SimpleGroupID(N) : RngIntElt -> Tup

 
Database of Small Groups

      Basic Small Group Functions
            SmallGroupDatabase() : -> DB
            delete D : DB ->
            SmallGroupDatabaseLimit() : -> RngIntElt
            IsInSmallGroupDatabase(o) : RngIntElt -> BoolElt
            NumberOfSmallGroups(o) : RngIntElt -> RngIntElt
            SmallGroup(o, n) : RngIntElt, RngIntElt -> Grp
            SmallGroup(o: parameters) : RngIntElt -> Grp
            SmallGroup(o, f: parameters) : RngIntElt, Program -> Grp
            IsSoluble(D, o, n) : DB, RngIntElt, RngIntElt -> Grp
            SmallGroupIsInsoluble(o, n) : RngIntElt, RngIntElt -> Grp
            SmallGroup(o, f: parameters) : RngIntElt, Program -> Grp
            SmallGroups(o: parameters) : RngIntElt -> [* Grp *]
            SmallGroups(S: parameters) : [RngIntElt] -> [* Grp *]
            SmallGroups(o, f: parameters) : RngIntElt, Program -> [* Grp *]
            SmallGroups(S, f: parameters) : [RngIntElt], Program -> [* Grp *]
            Example GrpData_SmallGroups (H73E1)

      Processes
            SmallGroupProcess(o: parameters) : RngIntElt -> Process
            SmallGroupProcess(S: parameters) : [RngIntElt] -> Process
            SmallGroupProcess(o, f: parameters) : RngIntElt, Program -> Process
            SmallGroupProcess(S, f: parameters) : [RngIntElt], Program -> Process
            IsEmpty(p) : Process -> BoolElt
            Current(p) : Process -> Grp
            CurrentLabel(p) : Process -> RngIntElt, RngIntElt
            Advance(~p) : Process ->
            Example GrpData_sg-process (H73E2)

      Small Group Identification
            IdentifyGroup(G): Grp -> Tup
            CanIdentifyGroup(o) : RngIntElt -> BoolElt
            Example GrpData_SmallIdentify (H73E3)
            Example GrpData_IdentifyGroup (H73E4)

      Accessing Internal Data
            Data(D, o, n) : DB, RngIntElt, RngIntElt -> List
            SmallGroupEncoding(G) : GrpPC -> RngIntElt, RngIntElt
            SmallGroupDecoding(c, o) : RngIntElt, RngIntElt -> GrpPC
            Example GrpData_SmallInternal (H73E5)

      Groups with Order Divisible by Only 4 Primes
            Group4P(o, i) : RngIntElt, RngIntElt -> Grp
            Groups4P(o) : RngIntElt -> List
            NumberOfGroups4P(o) : RngIntElt -> RngIntElt
            IdentifyGroup4P(G) : GrpPC -> RngIntElt

 
The p-groups of Order Dividing p⁷
      SearchPGroups(p, n: parameters) : RngIntElt, RngIntElt -> SeqEnum
      CountPGroups(p, n: parameters) : RngIntElt, RngIntElt -> SeqEnum
      Example GrpData_p7 (H73E6)

 
Metacyclic p-groups
      MetacyclicPGroups(p, n: parameters) : RngIntElt, RngIntElt -> SeqEnum
      IsMetacyclicPGroup(P) : Grp -> BoolElt
      InvariantsMetacyclicPGroup(P) : Grp -> Tup
      StandardMetacyclicPGroup(P): Grp -> GrpPC
      NumberOfMetacyclicPGroups(p, n): RngIntElt, RngIntElt -> SeqEnum
      HasAllPQuotientsMetacyclic(G): GrpFP -> BoolElt, SeqEnum
      Example GrpData_meta (H73E7)

 
Database of Perfect Groups

      Specifying an Entry of the Database

      Creating the Database
            PerfectGroupDatabase() : -> DB

      Accessing the Database
            Group(D, i): DB, RngIntElt -> GrpFP, SeqEnum
            IdentificationNumber(D, i): DB, RngIntElt -> RngIntElt
            NumberOfRepresentations(D, i): DB, RngIntElt -> RngIntElt
            PermutationRepresentation(D, i: parameters): DB, RngIntElt -> Hom(Grp), GrpFP, GrpPerm
            PermutationGroup(D, i: parameters): DB, RngIntElt -> GrpPerm

      Finding Legal Keys
            # D : DB -> RngIntElt
            NumberOfGroups(D, o) : DB, RngIntElt -> RngIntElt
            TopQuotients(D) : DB -> SetIndx
            ExtensionPrimes(D, Q) : DB, MonStgElt -> SetEnum
            ExtensionExponents(D, Q, p) : DB, MonStgElt, RngIntElt -> SetEnum
            ExtensionNumbers(D, Q, p, r) : DB, MonStgElt, RngIntElt, RngIntElt -> SetEnum
            ExtensionClasses(D, Q) : DB, MonStgElt -> SetEnum
            Example GrpData_perfgps (H73E8)

 
Database of Almost-Simple Groups

      The Record Fields

      Creating the Database
            AlmostSimpleGroupDatabase() : -> DB

      Accessing the Database
            # D : DB -> RngIntElt
            GroupData(D, i): DB, RngIntElt -> Rec
            ExistsGroupData(D, o1, o2): DB, RngIntElt, RngIntElt -> BoolElt
            NumberOfGroups(D, o1, o2): DB, RngIntElt, RngIntElt -> RngIntElt, RngIntElt
            IdentifyAlmostSimpleGroup(G) : GrpPerm -> Map, GrpPerm
            Example GrpData_sgdb (H73E9)

 
Database of Transitive Groups

      Accessing the Databases
            TransitiveGroupDatabaseLimit() : -> RngIntElt
            NumberOfTransitiveGroups(d) : RngIntElt -> RngIntElt
            TransitiveGroup(d, n) : RngIntElt, RngIntElt -> GrpPerm, MonStgElt
            TransitiveGroupDescription(d, n) : RngIntElt, RngIntElt -> MonStgElt
            TransitiveGroupDescription(G) : GrpPerm -> MonStgElt
            TransitiveGroup(d) : RngIntElt -> GrpPerm, MonStgElt
            TransitiveGroup(d, f) : RngIntElt, Program -> GrpPerm, MonStgElt
            TransitiveGroup(S, f) : [RngIntElt], Program -> GrpPerm, MonStgElt
            TransitiveGroups(d: parameters) : RngIntElt -> [GrpPerm]
            TransitiveGroups(S: parameters) : [RngIntElt] -> [GrpPerm]
            TransitiveGroups(d, f) : RngIntElt, Program -> [GrpPerm]
            TransitiveGroups(S, f) : [RngIntElt], Program -> [GrpPerm]
            Example GrpData_Transitive (H73E10)

      Processes
            TransitiveGroupProcess(d) : RngIntElt -> Process
            TransitiveGroupProcess(S) : [RngIntElt] -> Process
            TransitiveGroupProcess(d, f) : RngIntElt, Program -> Process
            TransitiveGroupProcess(S, f) : [RngIntElt], Program -> Process
            IsEmpty(p) : Process -> BoolElt
            Current(p) : Process -> GrpPerm, MonStgElt
            CurrentLabel(p) : Process -> RngIntElt, RngIntElt
            Advance(~p) : Process ->
            Example GrpData_TransitiveProcess (H73E11)

      Transitive Group Identification
            TransitiveGroupIdentification(G) : GrpPerm -> RngIntElt, RngIntElt
            Example GrpData_TransitiveId (H73E12)

 
Database of Primitive Groups

      Accessing the Databases
            PrimitiveGroupDatabaseLimit() : -> RngIntElt
            NumberOfPrimitiveGroups(d) : RngIntElt -> RngIntElt
            PrimitiveGroup(d, n) : RngIntElt, RngIntElt -> GrpPerm, MonStgElt, MonStgElt
            PrimitiveGroupDescription(d, n) : RngIntElt, RngIntElt -> MonStgElt
            PrimitiveGroup(d) : RngIntElt -> GrpPerm, MonStgElt, MonStgElt
            PrimitiveGroup(d, f) : RngIntElt, Program -> GrpPerm, MonStgElt
            PrimitiveGroup(S, f) : [RngIntElt], Program -> GrpPerm, MonStgElt
            PrimitiveGroups(d: parameters) : RngIntElt -> [GrpPerm]
            PrimitiveGroups(S: parameters) : [RngIntElt] -> [GrpPerm]
            PrimitiveGroups(d, f: parameters) : RngIntElt, Program -> [GrpPerm]
            Example GrpData_Primitive (H73E13)

      Processes
            PrimitiveGroupProcess(d: parameters) : RngIntElt -> Process
            PrimitiveGroupProcess(d, f: parameters) : RngIntElt, Program -> Process
            IsEmpty(p) : Process -> BoolElt
            Current(p) : Process -> GrpPerm, MonStgElt
            CurrentLabel(p) : Process -> RngIntElt, RngIntElt
            Advance(~p) : Process ->
            Example GrpData_PrimitiveProcess (H73E14)

      Primitive Group Identification
            PrimitiveGroupIdentification(G) : GrpPerm -> RngIntElt, RngIntElt
            Example GrpData_PrimitiveId (H73E15)

 
Database of Rational Maximal Finite Matrix Groups
      RationalMatrixGroupDatabase() : -> DB
      LargestDimension(D) : DB -> RngIntElt
      # D : DB -> RngIntElt
      NumberOfGroups(D, d) : DB, RngIntElt -> RngIntElt
      Group(D, i): DB, RngIntElt -> GrpMat
      Lattice(D, i): DB, RngIntElt -> Lat
      Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
      Lattice(D, d, i): DB, RngIntElt, RngIntElt -> Lat
      Example GrpData_ratgps1 (H73E16)

 
Database of Integral Maximal Finite Matrix Groups
      IntegralMatrixGroupDatabase() : -> DB
      LargestDimension(D) : DB -> RngIntElt
      # D : DB -> RngIntElt
      NumberOfGroups(D, d) : DB, RngIntElt -> RngIntElt
      Group(D, i): DB, RngIntElt -> GrpMat
      Lattice(D, i): DB, RngIntElt -> Lat, SeqEnum
      Construction(D, i): DB, RngIntElt -> MonStgElt, SeqEnum
      Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
      Lattice(D, d, i): DB, RngIntElt, RngIntElt -> Lat, SeqEnum
      Construction(D, d, i): DB, RngIntElt, RngIntElt -> MonStgElt, SeqEnum
      Example GrpData_Integral (H73E17)

 
Database of Finite Quaternionic Matrix Groups
      QuaternionicMatrixGroupDatabase() : -> DB
      LargestDimension(D) : DB -> RngIntElt
      # D : DB -> RngIntElt
      NumberOfGroups(D, d) : DB, RngIntElt -> RngIntElt
      Group(D, i): DB, RngIntElt -> GrpMat
      Lattice(D, i): DB, RngIntElt -> Lat, SeqEnum
      Construction(D, i): DB, RngIntElt -> MonStgElt, RngIntElt
      Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
      Lattice(D, d, i): DB, RngIntElt, RngIntElt -> Lat, SeqEnum
      Construction(D, d, i): DB, RngIntElt, RngIntElt -> MonStgElt, RngIntElt
      Example GrpData_Quaternionic (H73E18)

 
Database of Finite Symplectic Matrix Groups
      SymplecticMatrixGroupDatabase() : -> DB
      LargestDimension(D) : DB -> RngIntElt
      # D : DB -> RngIntElt
      NumberOfGroups(D, d) : DB, RngIntElt -> RngIntElt
      Group(D, i): DB, RngIntElt -> GrpMat
      Lattice(D, i): DB, RngIntElt -> Lat, SeqEnum
      Construction(D, i): DB, RngIntElt -> MonStgElt
      Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
      Lattice(D, d, i): DB, RngIntElt, RngIntElt -> Lat, SeqEnum
      Construction(D, d, i): DB, RngIntElt, RngIntElt -> MonStgElt
      Example GrpData_Symplectic (H73E19)

 
Database of Irreducible Matrix Groups

      Accessing the Database
            NumberOfIrreducibleMatrixGroups(k, p) : RngIntElt, RngIntElt -> RngIntElt
            IrreducibleMatrixGroup(k, p, n) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
            Example GrpData_IrredMat (H73E20)

 
Database of Quasisimple Matrix Groups
      QuasisimpleMatrixGroup(N, d, p : parameters) : MonStgElt, RngIntElt, RngIntElt ->GrpMat
      QuasisimpleMatrixGroups(): -> SeqEnum

 
Database of Soluble Irreducible Groups

      Basic Functions
            IsolGroupDatabase() : -> DB
            IsolGroup(n, p, i) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
            IsolNumberOfDegreeField(n, p) : RngIntElt, RngIntElt -> RngIntElt
            IsolInfo(n, p, i) : RngIntElt, RngIntElt, RngIntElt -> MonStgElt
            IsolOrder(n, p, i) : RngIntElt, RngIntElt, RngIntElt -> RngIntElt
            IsolMinBlockSize(n, p, i) : RngIntElt, RngIntElt, RngIntElt -> RngIntElt
            IsolIsPrimitive(n, p, i) : RngIntElt, RngIntElt, RngIntElt -> BoolElt
            IsolGuardian(n, p, i) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
            Example GrpData_IsolGroup (H73E21)

      Searching with Predicates
            IsolGroupSatisfying(f) : Any -> GrpMat
            IsolGroupOfDegreeSatisfying(d, f) : RngIntElt, Any -> GrpMat
            IsolGroupOfDegreeFieldSatisfying(d, p, f) : RngIntElt, RngIntElt, Any -> GrpMat
            IsolGroupsSatisfying(f) : Any -> SeqEnum
            IsolGroupsOfDegreeSatisfying(d, f) : RngIntElt, Any -> SeqEnum
            IsolGroupsOfDegreeFieldSatisfying(d, p, f) : RngIntElt, RngIntElt, Any -> SeqEnum

      Associated Functions
            Getvecs(G) : GrpMat -> SeqEnum
            Semidir(G, Q) : GrpMat, SeqEnum -> GrpPerm

      Processes
            IsolProcess() : -> Process
            IsolProcessOfDegree(d) : . -> Process
            IsolProcessOfField(p) : . -> Process
            IsolProcessOfDegreeField(d, p) : ., . -> Process
            IsEmpty(p) : Process -> BoolElt
            Current(p) : Process -> GrpMat
            CurrentLabel(p) : Process -> RngIntElt, RngIntElt, RngIntElt
            Advance(~p) : Process ->
            Example GrpData_sg-process (H73E22)

 
Database of ATLAS Groups
      Example GrpData_ATLAS-names (H73E23)

      Accessing the Database
            ATLASGroupNames() : -> SetIndx[MonStgElt]
            ATLASGroup(N) : MonStgElt -> GrpAtlas

      Accessing the ATLAS Groups
            Order(A) : GrpAtlas -> RngIntElt
            Multiplier(A) : GrpAtlas -> RngIntElt
            MatRepKeys(A) : GrpAtlas -> SeqEnum[DBAtlasKeyMatRep]
            MatRepDegrees(A) : GrpAtlas -> SetEnum[RngIntElt]
            Degree(K) : DBAtlasKeyMatRep -> RngIntElt
            MatRepFieldSizes(A) : GrpAtlas -> SetEnum[RngIntElt]
            MatRepCharacteristics(A) : GrpAtlas -> SetEnum[RngIntElt]
            Field(K) : DBAtlasKeyMatRep -> FldFin
            PermRepKeys(A) : GrpAtlas -> SeqEnum[DBAtlasKeyPermRep]
            PermRepDegrees(A) : GrpAtlas -> SetEnum[RngIntElt]
            Degree(K) : DBAtlasKeyPermRep -> RngIntElt

      Representations of the ATLAS Groups
            MatrixGroup(K) : DBAtlasKeyMatRep -> GrpMat
            MatRep(K) : DBAtlasKeyMatRep -> SeqEnum[GrpMatElt]
            PermutationGroup(K) : DBAtlasKeyPermRep -> GrpPerm
            PermRep(K) : DBAtlasKeyPermRep -> SeqEnum[GrpPermElt]
            Example GrpData_J2 (H73E24)

 
Fundamental Groups of 3-Manifolds

      Basic Functions
            ManifoldDatabase() : -> DB
            Manifold(D, i) : DB, RngIntElt -> Rec

      Accessing the Data
            Example GrpData_manifolds (H73E25)

 
Automatic Groups of 3-Manifolds
      AutomaticGroup(i) : RngIntElt -> GrpAtc
      AutomaticGroupIndices() : -> [RngIntElt]
      AutomaticGroupNames() : -> [MonStgElt]

 
Bibliography

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