Databases of Groups
- Introduction
- Database of Simple Groups
- Database of Small Groups
- Basic Small Group Functions
SmallGroupDatabase() → DBOpenSmallGroupDatabase() → DBdelete D: DBSmallGroupDatabaseLimit() → RngIntEltSmallGroupDatabaseLimit(D): DB → RngIntEltIsInSmallGroupDatabase(o): RngIntElt → BoolEltIsInSmallGroupDatabase(D, o): DB, RngIntElt → BoolEltNumberOfSmallGroups(o): RngIntElt → RngIntEltNumberOfSmallGroups(D, o): DB, RngIntElt → RngIntEltSmallGroup(o, n): RngIntElt, RngIntElt → GrpSmallGroup(D, o, n): DB, RngIntElt, RngIntElt → GrpGroup(D, o, n): DB, RngIntElt, RngIntElt → GrpSmallGroup(o: parameters): RngIntElt → GrpSmallGroup(D, o: parameters): DB, RngIntElt → GrpSmallGroup(o, f: parameters): RngIntElt, Program → GrpSmallGroup(D, o, f: parameters): RngIntElt, Program → GrpIsSoluble(D, o, n): DB, RngIntElt, RngIntElt → GrpIsSolvable(D, o, n): DB, RngIntElt, RngIntElt → GrpSmallGroupIsSoluble(o, n): RngIntElt, RngIntElt → GrpSmallGroupIsSoluble(D, o, n): DB, RngIntElt, RngIntElt → GrpSmallGroupIsSolvable(o, n): RngIntElt, RngIntElt → GrpSmallGroupIsSolvable(D, o, n): DB, RngIntElt, RngIntElt → GrpSmallGroupIsInsoluble(o, n): RngIntElt, RngIntElt → GrpSmallGroupIsInsoluble(D, o, n): DB, RngIntElt, RngIntElt → GrpSmallGroupIsInsolvable(o, n): RngIntElt, RngIntElt → GrpSmallGroupIsInsolvable(D, o, n): DB, RngIntElt, RngIntElt → GrpSmallGroup(o, f: parameters): RngIntElt, Program → GrpSmallGroup(S, f: parameters): [RngIntElt], Program → GrpSmallGroup(D, o, f: parameters): DB, RngIntElt, Program → GrpSmallGroup(D, S, f: parameters): DB, [RngIntElt], Program → GrpSmallGroups(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: Small Groups
- Processes
- Small Group Identification
- Accessing Internal Data
- Groups with Order Divisible by Only 4 Primes
- The \(p\)-groups of Order Dividing \(p^7\)
- Metacyclic \(p\)-groups
- Database of Perfect Groups
- Specifying an Entry of the Database
- Creating the Database
- Accessing the Database
Group(D, i): DB, RngIntElt → GrpFP, SeqEnumGroup(D, o, i): DB, RngIntElt, RngIntElt → GrpFP, SeqEnumGroup(D, Q): DB, MonStgElt → GrpFP, SeqEnumGroup(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → GrpFP, SeqEnumIdentificationNumber(D, i): DB, RngIntElt → RngIntEltIdentificationNumber(D, o, i): DB, RngIntElt, RngIntElt → RngIntEltIdentificationNumber(D, Q): DB, MonStgElt → RngIntEltIdentificationNumber({D, Q, p, r, }{n: parameters}): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntEltNumberOfRepresentations(D, i): DB, RngIntElt → RngIntEltNumberOfRepresentations(D, o, i): DB, RngIntElt, RngIntElt → RngIntEltNumberOfRepresentations(D, Q): DB, MonStgElt → RngIntEltNumberOfRepresentations({D, Q, p, }{r, n: parameters}): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntEltPermutationRepresentation(D, i: parameters): DB, RngIntElt → Hom(Grp), GrpFP, GrpPermPermutationRepresentation({D, o, }{i: parameters}): DB, RngIntElt, RngIntElt → Hom(Grp), GrpFP, GrpPermPermutationRepresentation(D, Q: parameters): DB, MonStgElt → Hom(Grp), GrpFP, GrpPermPermutationRepresentation({D, Q, }{p, r, n: parameters}): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → Hom(Grp), GrpFP, GrpPermPermutationGroup(D, i: parameters): DB, RngIntElt → GrpPermPermutationGroup(D, o, i: parameters): DB, RngIntElt, RngIntElt → GrpPermPermutationGroup(D, Q: parameters): DB, MonStgElt → GrpPermPermutationGroup(D, Q, p, r, n: parameters): DB, MonStgElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → GrpPerm
- Finding Legal Keys
# D: DB → RngIntEltNumberOfGroups(D): DB → RngIntEltNumberOfGroups(D, o): DB, RngIntElt → RngIntEltTopQuotients(D): DB → SetIndxExtensionPrimes(D, Q): DB, MonStgElt → SetEnumExtensionExponents(D, Q, p): DB, MonStgElt, RngIntElt → SetEnumExtensionNumbers(D, Q, p, r): DB, MonStgElt, RngIntElt, RngIntElt → SetEnumExtensionClasses(D, Q): DB, MonStgElt → SetEnumExample: perfgps
- Database of Almost-Simple Groups
- The Record Fields
- Creating the Database
- Accessing the Database
# D: DB → RngIntEltGroupData(D, i): DB, RngIntElt → RecGroupData(D, o1, o2, k): DB, RngIntElt, RngIntElt, RngIntElt → RecExistsGroupData(D, o1, o2): DB, RngIntElt, RngIntElt → BoolEltExistsGroupData(D, o1, o2, i): DB, RngIntElt, RngIntElt, RngIntElt → BoolElt, RecNumberOfGroups(D, o1, o2): DB, RngIntElt, RngIntElt → RngIntElt, RngIntEltIdentifyAlmostSimpleGroup(G): GrpPerm → Map, GrpPermIdentifyAlmostSimpleGroup(G): GrpMat → Map, GrpPermExample: sgdb
- Database of Transitive Groups
- Accessing the Databases
TransitiveGroupDatabaseLimit() → RngIntEltNumberOfTransitiveGroups(d): RngIntElt → RngIntEltTransitiveGroup(d, n): RngIntElt, RngIntElt → GrpPerm, MonStgEltTransitiveGroupDescription(d, n): RngIntElt, RngIntElt → MonStgEltTransitiveGroupDescription(G): GrpPerm → MonStgEltTransitiveGroup(d): RngIntElt → GrpPerm, MonStgEltTransitiveGroup(d, f): RngIntElt, Program → GrpPerm, MonStgEltTransitiveGroup(S, f): [RngIntElt], Program → GrpPerm, MonStgEltTransitiveGroups(d: parameters): RngIntElt → [GrpPerm]TransitiveGroups(S: parameters): [RngIntElt] → [GrpPerm]TransitiveGroups(d, f): RngIntElt, Program → [GrpPerm]TransitiveGroups(S, f): [RngIntElt], Program → [GrpPerm]Example: TransitiveExample: Non Conj
- Processes
- Transitive Group Identification
- Database of Primitive Groups
- Accessing the Databases
PrimitiveGroupDatabaseLimit() → RngIntEltNumberOfPrimitiveGroups(d): RngIntElt → RngIntEltNumberOfPrimitiveSolubleGroups(d): RngIntElt → RngIntEltNumberOfPrimitiveAffineGroups(d): RngIntElt → RngIntEltNumberOfPrimitiveDiagonalGroups(d): RngIntElt → RngIntEltNumberOfPrimitiveProductGroups(d): RngIntElt → RngIntEltNumberOfPrimitiveAlmostSimpleGroups(d): RngIntElt → RngIntEltPrimitiveGroup(d, n): RngIntElt, RngIntElt → GrpPerm, MonStgElt, MonStgEltPrimitiveGroupDescription(d, n): RngIntElt, RngIntElt → MonStgEltPrimitiveGroup(d): RngIntElt → GrpPerm, MonStgElt, MonStgEltPrimitiveGroup(d, f): RngIntElt, Program → GrpPerm, MonStgEltPrimitiveGroup(S, f): [RngIntElt], Program → GrpPerm, MonStgEltPrimitiveGroups(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
- Processes
- Primitive Group Identification
- Database of Rational Maximal Finite Matrix Groups
RationalMatrixGroupDatabase() → DBLargestDimension(D): DB → RngIntElt# D: DB → RngIntEltNumberOfGroups(D): DB → RngIntEltNumberOfLattices(D): DB → RngIntEltNumberOfGroups(D, d): DB, RngIntElt → RngIntEltNumberOfLattices(D, d): DB, RngIntElt → RngIntEltGroup(D, i): DB, RngIntElt → GrpMatLattice(D, i): DB, RngIntElt → LatGroup(D, d, i): DB, RngIntElt, RngIntElt → GrpMatLattice(D, d, i): DB, RngIntElt, RngIntElt → LatExample: ratgps1
- Database of Integral Maximal Finite Matrix Groups
IntegralMatrixGroupDatabase() → DBLargestDimension(D): DB → RngIntElt# D: DB → RngIntEltNumberOfGroups(D): DB → RngIntEltNumberOfLattices(D): DB → RngIntEltNumberOfGroups(D, d): DB, RngIntElt → RngIntEltNumberOfLattices(D, d): DB, RngIntElt → RngIntEltGroup(D, i): DB, RngIntElt → GrpMatLattice(D, i): DB, RngIntElt → Lat, SeqEnumConstruction(D, i): DB, RngIntElt → MonStgElt, SeqEnumGroup(D, d, i): DB, RngIntElt, RngIntElt → GrpMatLattice(D, d, i): DB, RngIntElt, RngIntElt → Lat, SeqEnumConstruction(D, d, i): DB, RngIntElt, RngIntElt → MonStgElt, SeqEnumExample: Integral
- Database of Finite Quaternionic Matrix Groups
QuaternionicMatrixGroupDatabase() → DBLargestDimension(D): DB → RngIntElt# D: DB → RngIntEltNumberOfGroups(D): DB → RngIntEltNumberOfLattices(D): DB → RngIntEltNumberOfGroups(D, d): DB, RngIntElt → RngIntEltNumberOfLattices(D, d): DB, RngIntElt → RngIntEltGroup(D, i): DB, RngIntElt → GrpMatLattice(D, i): DB, RngIntElt → Lat, SeqEnumConstruction(D, i): DB, RngIntElt → MonStgElt, RngIntEltGroup(D, d, i): DB, RngIntElt, RngIntElt → GrpMatLattice(D, d, i): DB, RngIntElt, RngIntElt → Lat, SeqEnumConstruction(D, d, i): DB, RngIntElt, RngIntElt → MonStgElt, RngIntEltExample: Quaternionic
- Database of Finite Symplectic Matrix Groups
SymplecticMatrixGroupDatabase() → DBLargestDimension(D): DB → RngIntElt# D: DB → RngIntEltNumberOfGroups(D): DB → RngIntEltNumberOfLattices(D): DB → RngIntEltNumberOfGroups(D, d): DB, RngIntElt → RngIntEltNumberOfLattices(D, d): DB, RngIntElt → RngIntEltGroup(D, i): DB, RngIntElt → GrpMatLattice(D, i): DB, RngIntElt → Lat, SeqEnumConstruction(D, i): DB, RngIntElt → MonStgEltGroup(D, d, i): DB, RngIntElt, RngIntElt → GrpMatLattice(D, d, i): DB, RngIntElt, RngIntElt → Lat, SeqEnumConstruction(D, d, i): DB, RngIntElt, RngIntElt → MonStgEltExample: Symplectic
- Database of Irreducible Matrix Groups
- Database of Quasisimple Matrix Groups
- Database of Soluble Irreducible Groups
- Basic Functions
IsolGroupDatabase() → DBIsolGroup(n, p, i): RngIntElt, RngIntElt, RngIntElt → GrpMatGroup(D, n, p, i): DB, RngIntElt, RngIntElt, RngIntElt → GrpMatIsolNumberOfDegreeField(n, p): RngIntElt, RngIntElt → RngIntEltIsolInfo(n, p, i): RngIntElt, RngIntElt, RngIntElt → MonStgEltIsolOrder(n, p, i): RngIntElt, RngIntElt, RngIntElt → RngIntEltIsolMinBlockSize(n, p, i): RngIntElt, RngIntElt, RngIntElt → RngIntEltIsolIsPrimitive(n, p, i): RngIntElt, RngIntElt, RngIntElt → BoolEltIsolGuardian(n, p, i): RngIntElt, RngIntElt, RngIntElt → GrpMatExample: Isol Group
- Searching with Predicates
IsolGroupSatisfying(f): Any → GrpMatIsolGroupOfDegreeSatisfying(d, f): RngIntElt, Any → GrpMatIsolGroupOfDegreeFieldSatisfying(d, p, f): RngIntElt, RngIntElt, Any → GrpMatIsolGroupsSatisfying(f): Any → SeqEnumIsolGroupsOfDegreeSatisfying(d, f): RngIntElt, Any → SeqEnumIsolGroupsOfDegreeFieldSatisfying(d, p, f): RngIntElt, RngIntElt, Any → SeqEnum
- Associated Functions
- Processes
- Database of ATLAS Groups
- Fundamental Groups of 3-Manifolds
- Automatic Groups of 3-Manifolds
