Databases of Groups

• Introduction
• Database of Simple Groups
  • SimpleGroup(N): RngIntElt → Grp
  • SimpleGroup(S): MonStgElt → Grp
  • SimpleGroup(T): Tup → Grp
  • SimpleGroupOfOrder(M): RngIntElt → Grp
  • SimpleGroup(F): RngIntEltFac → 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
    • OpenSmallGroupDatabase() → DB
    • delete D: DB
    • SmallGroupDatabaseLimit() → RngIntElt
    • SmallGroupDatabaseLimit(D): DB → RngIntElt
    • IsInSmallGroupDatabase(o): RngIntElt → BoolElt
    • IsInSmallGroupDatabase(D, o): DB, RngIntElt → BoolElt
    • NumberOfSmallGroups(o): RngIntElt → RngIntElt
    • NumberOfSmallGroups(D, o): DB, RngIntElt → RngIntElt
    • SmallGroup(o, n): RngIntElt, RngIntElt → Grp
    • SmallGroup(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • Group(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • SmallGroup(o: parameters): RngIntElt → Grp
    • SmallGroup(D, o: parameters): DB, RngIntElt → Grp
    • SmallGroup(o, f: parameters): RngIntElt, Program → Grp
    • SmallGroup(D, o, f: parameters): RngIntElt, Program → Grp
    • IsSoluble(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • IsSolvable(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • SmallGroupIsSoluble(o, n): RngIntElt, RngIntElt → Grp
    • SmallGroupIsSoluble(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • SmallGroupIsSolvable(o, n): RngIntElt, RngIntElt → Grp
    • SmallGroupIsSolvable(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • SmallGroupIsInsoluble(o, n): RngIntElt, RngIntElt → Grp
    • SmallGroupIsInsoluble(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • SmallGroupIsInsolvable(o, n): RngIntElt, RngIntElt → Grp
    • SmallGroupIsInsolvable(D, o, n): DB, RngIntElt, RngIntElt → Grp
    • SmallGroup(o, f: parameters): RngIntElt, Program → Grp
    • SmallGroup(S, f: parameters): [RngIntElt], Program → Grp
    • SmallGroup(D, o, f: parameters): DB, RngIntElt, Program → Grp
    • SmallGroup(D, S, f: parameters): DB, [RngIntElt], Program → Grp
    • SmallGroups(o: parameters): RngIntElt → [* Grp *]
    • SmallGroups(D, o: parameters): DB, RngIntElt → [* Grp *]
    • SmallGroups(S: parameters): [RngIntElt] → [* Grp *]
    • SmallGroups(D, S: parameters): DB, [RngIntElt] → [* Grp *]
    • SmallGroups(o, f: parameters): RngIntElt, Program → [* Grp *]
    • SmallGroups(D, o, f: parameters): DB, RngIntElt, Program → [* Grp *]
    • SmallGroups(S, f: parameters): [RngIntElt], Program → [* Grp *]
    • SmallGroups(D, S, f: parameters): DB, [RngIntElt], Program → [* Grp *]
    • Example: Non-Abelian Groups of Order 27
  • 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: Nilpotency Classes of Groups of Order 128
  • Small Group Identification
    • IdentifyGroup(G): Grp → Tup
    • IdentifyGroup(D, G): DB, Grp → Tup
    • CanIdentifyGroup(o): RngIntElt → BoolElt
    • CanIdentifyGroup(D, o): DB, RngIntElt → BoolElt
    • IdentifyGroupBySearch(G): Grp → Tup
    • Example: Identifying a Small Group
    • Example: Identifying a Group by Search
    • Example: Identifying a Finitely Presented Group
  • Accessing Internal Data
    • Data(D, o, n): DB, RngIntElt, RngIntElt → List
    • SmallGroupEncoding(G): GrpPC → RngIntElt, RngIntElt
    • SmallGroupDecoding(c, o): RngIntElt, RngIntElt → GrpPC
    • Example: Internal Data of the Small Groups Database
  • 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
    • IdentifyGroup4P(G): GrpPerm → RngIntElt
    • IdentifyGroup4P(G): GrpMat → RngIntElt
• The \(p\)-groups of Order Dividing \(p^7\)
  • SearchPGroups(p, n: parameters): RngIntElt, RngIntElt → SeqEnum
  • CountPGroups(p, n: parameters): RngIntElt, RngIntElt → SeqEnum
  • Example: Searching Groups of Order 19^7
• 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
  • HasAllPQuotientsMetacyclic(G, p): GrpFP → BoolElt
  • Example: Metacyclic 3-Groups of Order 729
• 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
    • Group(D, o, i): DB, RngIntElt, RngIntElt → GrpFP, SeqEnum
    • Group(D, Q): DB, MonStgElt → GrpFP, SeqEnum
    • Group(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → GrpFP, SeqEnum
    • IdentificationNumber(D, i): DB, RngIntElt → RngIntElt
    • IdentificationNumber(D, o, i): DB, RngIntElt, RngIntElt → RngIntElt
    • IdentificationNumber(D, Q): DB, MonStgElt → RngIntElt
    • IdentificationNumber(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt
    • NumberOfRepresentations(D, i): DB, RngIntElt → RngIntElt
    • NumberOfRepresentations(D, o, i): DB, RngIntElt, RngIntElt → RngIntElt
    • NumberOfRepresentations(D, Q): DB, MonStgElt → RngIntElt
    • NumberOfRepresentations(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt
    • PermutationRepresentation(D, i: parameters): DB, RngIntElt → Hom(Grp), GrpFP, GrpPerm
    • PermutationRepresentation(D, o, i: parameters): DB, RngIntElt, RngIntElt → Hom(Grp), GrpFP, GrpPerm
    • PermutationRepresentation(D, Q: parameters): DB, MonStgElt → Hom(Grp), GrpFP, GrpPerm
    • PermutationRepresentation(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → Hom(Grp), GrpFP, GrpPerm
    • PermutationGroup(D, i: parameters): DB, RngIntElt → GrpPerm
    • PermutationGroup(D, o, i: parameters): DB, RngIntElt, RngIntElt → GrpPerm
    • PermutationGroup(D, Q: parameters): DB, MonStgElt → GrpPerm
    • PermutationGroup(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → GrpPerm
  • Finding Legal Keys
    • # D: DB → RngIntElt
    • NumberOfGroups(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: Finding an Extension of L(3, 4)
• 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
    • GroupData(D, o1, o2, k): DB, RngIntElt, RngIntElt, RngIntElt → Rec
    • ExistsGroupData(D, o1, o2): DB, RngIntElt, RngIntElt → BoolElt
    • ExistsGroupData(D, o1, o2, i): DB, RngIntElt, RngIntElt, RngIntElt → BoolElt, Rec
    • NumberOfGroups(D, o1, o2): DB, RngIntElt, RngIntElt → RngIntElt, RngIntElt
    • IdentifyAlmostSimpleGroup(G): GrpPerm → Map, GrpPerm
    • IdentifyAlmostSimpleGroup(G): GrpMat → Map, GrpPerm
    • Example: Querying an Almost Simple Group of Order 720
• 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: Transitive Groups of Degree 8
    • Example: Isomorphic Non-Conjugate Transitive Groups
  • 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: Processing Transitive Groups of Degree 5
  • Transitive Group Identification
    • TransitiveGroupIdentification(G): GrpPerm → RngIntElt, RngIntElt
    • Example: Identifying a Transitive Group
• Database of Primitive Groups
  • Accessing the Databases
    • PrimitiveGroupDatabaseLimit() → RngIntElt
    • NumberOfPrimitiveGroups(d): RngIntElt → RngIntElt
    • NumberOfPrimitiveSolubleGroups(d): RngIntElt → RngIntElt
    • NumberOfPrimitiveAffineGroups(d): RngIntElt → RngIntElt
    • NumberOfPrimitiveDiagonalGroups(d): RngIntElt → RngIntElt
    • NumberOfPrimitiveProductGroups(d): RngIntElt → RngIntElt
    • NumberOfPrimitiveAlmostSimpleGroups(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( : parameters) → [GrpPerm]
    • PrimitiveGroups(d, f: parameters): RngIntElt, Program → [GrpPerm]
    • PrimitiveGroups(S, f): [RngIntElt], Program → [GrpPerm]
    • PrimitiveGroups(f): Program → [GrpPerm]
    • Example: Primitive Groups of Degree 625
  • Processes
    • PrimitiveGroupProcess(d: parameters): RngIntElt → Process
    • PrimitiveGroupProcess(S: parameters): [RngIntElt] → Process
    • PrimitiveGroupProcess(: parameters) → Process
    • PrimitiveGroupProcess(d, f: parameters): RngIntElt, Program → Process
    • PrimitiveGroupProcess(S, f: parameters): [RngIntElt], Program → Process
    • PrimitiveGroupProcess(f: parameters): Program → Process
    • IsEmpty(p): Process → BoolElt
    • Current(p): Process → GrpPerm, MonStgElt
    • CurrentLabel(p): Process → RngIntElt, RngIntElt
    • Advance(~p): Process
    • Example: Processing Primitive Groups of Degree 60
  • Primitive Group Identification
    • PrimitiveGroupIdentification(G): GrpPerm → RngIntElt, RngIntElt
    • Example: Identifying a Primitive Group
• Database of Rational Maximal Finite Matrix Groups
  • RationalMatrixGroupDatabase() → DB
  • LargestDimension(D): DB → RngIntElt
  • # D: DB → RngIntElt
  • NumberOfGroups(D): DB → RngIntElt
  • NumberOfLattices(D): DB → RngIntElt
  • NumberOfGroups(D, d): DB, RngIntElt → RngIntElt
  • NumberOfLattices(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: Counting Rational Matrix Groups
• Database of Integral Maximal Finite Matrix Groups
  • IntegralMatrixGroupDatabase() → DB
  • LargestDimension(D): DB → RngIntElt
  • # D: DB → RngIntElt
  • NumberOfGroups(D): DB → RngIntElt
  • NumberOfLattices(D): DB → RngIntElt
  • NumberOfGroups(D, d): DB, RngIntElt → RngIntElt
  • NumberOfLattices(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: Accessing Integral Matrix Groups
• Database of Finite Quaternionic Matrix Groups
  • QuaternionicMatrixGroupDatabase() → DB
  • LargestDimension(D): DB → RngIntElt
  • # D: DB → RngIntElt
  • NumberOfGroups(D): DB → RngIntElt
  • NumberOfLattices(D): DB → RngIntElt
  • NumberOfGroups(D, d): DB, RngIntElt → RngIntElt
  • NumberOfLattices(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: Quaternionic Matrix Groups of Dimension 36
• Database of Finite Symplectic Matrix Groups
  • SymplecticMatrixGroupDatabase() → DB
  • LargestDimension(D): DB → RngIntElt
  • # D: DB → RngIntElt
  • NumberOfGroups(D): DB → RngIntElt
  • NumberOfLattices(D): DB → RngIntElt
  • NumberOfGroups(D, d): DB, RngIntElt → RngIntElt
  • NumberOfLattices(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: Symplectic Matrix Groups of Dimension 16
• Database of Irreducible Matrix Groups
  • Accessing the Database
    • NumberOfIrreducibleMatrixGroups(k, p): RngIntElt, RngIntElt → RngIntElt
    • NumberOfSolubleIrreducibleMatrixGroups(k, p): RngIntElt, RngIntElt → RngIntElt
    • IrreducibleMatrixGroup(k, p, n): RngIntElt, RngIntElt, RngIntElt → GrpMat
    • Example: Irreducible Matrix Groups in GL(4, 5)
• 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
    • Group(D, n, p, i): DB, 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: Group of Degree 3 and Its Guardian
  • 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: Processing All Groups of Degree 3
• Database of ATLAS Groups
  • Example: Listing ATLAS Group Names
  • Accessing the Database
    • ATLASGroupNames() → SetIndx[MonStgElt]
    • ATLASGroup(N): MonStgElt → GrpAtlas
  • Accessing the ATLAS Groups
    • Order(A): GrpAtlas → RngIntElt
    • # G: 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: Representations of 2.J2.2
• Fundamental Groups of 3-Manifolds
  • Basic Functions
    • ManifoldDatabase() → DB
    • Manifold(D, i): DB, RngIntElt → Rec
    • Manifold(D, name): DB, MonStgElt → Rec
  • Accessing the Data
    • Example: Records from the 3-Manifolds Database
• Automatic Groups of 3-Manifolds
  • AutomaticGroup(i): RngIntElt → GrpAtc
  • AutomaticGroup(name): MonStgElt → GrpAtc
  • AutomaticGroup(r): Rec → GrpAtc
  • AutomaticGroupIndices() → [RngIntElt]
  • AutomaticGroupNames() → [MonStgElt]
• Bibliography