DATABASES OF GROUPS

DETAILS

 
Introduction

 
The Database of Small Groups

      Basic Small Group Functions
            SmallGroupDatabase() : -> DB
            delete D : DB; -> Nil
            SmallGroupDatabaseLimit() : -> RngIntElt
            IsInSmallGroupDatabase(o) : RngIntElt -> RngIntElt
            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 (H22E1)

      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 (H22E2)

      Small Group Identification
            IdentifyGroup(G): Grp -> Tup
            Example GrpData_SmallIdentify (H22E3)

      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 (H22E4)

 
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 -> GrpFP

      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 (H22E5)

 
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 -> GrpMat
      Example GrpData_ratgps1 (H22E6)

 
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, Rng
      Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
      Lattice(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
      Construction(D, d, i): DB, RngIntElt, RngIntElt -> MonStgElt, Rng
      Example GrpData_Quaternionic (H22E7)

 
Permutation Group Databases

      Accessing the Databases
            TransitiveGroupDatabaseLimit() : -> RngIntElt
            NumberOfTransitiveGroups(d) : RngIntElt -> RngIntElt
            TransitiveGroup(d, n) : RngIntElt, RngIntElt -> GrpPerm, MonStgElt
            TransitiveGroupDescription(d, n) : RngIntElt, RngIntElt -> 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 (H22E8)

      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 (H22E9)

      Transitive Group Identification
            TransitiveGroupIdentification(G) : GrpPerm -> RngIntElt, RngIntElt
            Example GrpData_TransitiveId (H22E10)

 
Database of Almost-Simple Groups

      The Fields of the Record

      Creating the Database
            AlmostSimpleGroupDatabase() : -> DB

      Accessing the Database
            # D : DB -> RngIntElt
            GroupData(D, i): DB, RngIntElt -> Rec
            ExistsGroupData(D, o1, o2): DB, RngIntElt, RngIntElt -> bool
            IdentifyAlmostSimpleGroup(G) : GrpPerm -> Map, GrpPerm
            Example GrpData_sgdb (H22E11)

 
The Database of Irreducible Soluble Matrix 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 (H22E12)

      Searching with predicates
            IsolGroupSatisfying(f) : Predicate -> GrpMat
            IsolGroupOfDegreeSatisfying(d, f) : RngIntElt, Predicate -> GrpMat
            IsolGroupOfDegreeFieldSatisfying(d,{p, f}) : RngIntElt, RngIntElt, Predicate -> GrpMat
            IsolGroupsSatisfying(f) : Predicate -> SeqEnum
            IsolGroupsOfDegreeSatisfying(d, f) : RngIntElt, Predicate -> SeqEnum
            IsolGroupsOfDegreeFieldSatisfying(d,{p, f}) : RngIntElt, RngIntElt, Predicate -> 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 ->

 
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