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)
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
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)
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)
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
Wed Apr 24 15:09:57 EST 2013