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DATABASES OF GROUPS

 
Acknowledgements
 
Introduction
 
Database of Small Groups
      Basic Small Group Functions
      Processes
      Small Group Identification
      Accessing Internal Data
 
The p-groups of Order Dividing p7
 
Metacyclic p-groups
 
Database of Perfect Groups
      Specifying an Entry of the Database
      Creating the Database
      Accessing the Database
      Finding Legal Keys
 
Database of Almost-Simple Groups
      The Record Fields
      Creating the Database
      Accessing the Database
 
Database of Transitive Groups
      Accessing the Databases
      Processes
      Transitive Group Identification
 
Database of Primitive Groups
      Accessing the Databases
      Processes
      Primitive Group Identification
 
Database of Rational Maximal Finite Matrix Groups
 
Database of Integral Maximal Finite Matrix Groups
 
Database of Finite Quaternionic Matrix Groups
 
Database of Finite Symplectic Matrix Groups
 
Database of Irreducible Matrix Groups
      Accessing the Database
 
Database of Quasisimple Matrix Groups
 
Database of Soluble Irreducible Groups
      Basic Functions
      Searching with Predicates
      Associated Functions
      Processes
 
Database of ATLAS Groups
      Accessing the Database
      Accessing the ATLAS Groups
      Representations of the ATLAS Groups
 
Fundamental Groups of 3-Manifolds
      Basic Functions
      Accessing the Data
 
Bibliography







DETAILS

 
Introduction

 
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 (H66E1)

      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 (H66E2)

      Small Group Identification
            IdentifyGroup(G): Grp -> Tup
            CanIdentifyGroup(o) : RngIntElt -> BoolElt
            Example GrpData_SmallIdentify (H66E3)

      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 (H66E4)

 
The p-groups of Order Dividing p7
      SearchPGroups(p, n: parameters) : RngIntElt, RngIntElt -> SeqEnum
      CountPGroups(p, n: parameters) : RngIntElt, RngIntElt -> SeqEnum
      Example GrpData_p7 (H66E5)

 
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 (H66E6)

 
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 (H66E7)

 
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 (H66E8)

 
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 (H66E9)

      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 (H66E10)

      Transitive Group Identification
            TransitiveGroupIdentification(G) : GrpPerm -> RngIntElt, RngIntElt
            Example GrpData_TransitiveId (H66E11)

 
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 (H66E12)

      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 (H66E13)

      Primitive Group Identification
            PrimitiveGroupIdentification(G) : GrpPerm -> RngIntElt, RngIntElt
            Example GrpData_PrimitiveId (H66E14)

 
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 (H66E15)

 
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 (H66E16)

 
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 (H66E17)

 
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 (H66E18)

 
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 (H66E19)

 
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 (H66E20)

      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 (H66E21)

 
Database of ATLAS Groups
      Example GrpData_ATLAS-names (H66E22)

      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]
            MatRepFieldSizes(A) : GrpAtlas -> SetEnum[RngIntElt]
            MatRepCharacteristics(A) : GrpAtlas -> SetEnum[RngIntElt]
            PermRepKeys(A) : GrpAtlas -> SeqEnum[DBAtlasKeyPermRep]
            PermRepDegrees(A) : GrpAtlas -> SetEnum[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 (H66E23)

 
Fundamental Groups of 3-Manifolds

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

      Accessing the Data
            Example GrpData_manifolds (H66E24)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012