V;*      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy z { | } ~  5Type for permutations, considered as group elements. Vx .^ g returns the image of a vertex or point x under the action of the permutation g .Construct a permutation from a list of cycles 'A trick: g^-1 returns the inverse of g Eb -^ g returns the image of an edge or block b under the action of g /Generators for Cn, the cyclic group of order n /Generators for Sn, the symmetric group on [1..n] 1Generators for An, the alternating group on [1..n] RGiven generators for a group, return a (sorted) list of all elements of the group F      ! 3 "#$%&'()*+,-. /0123456789:;<=>?@ABCDEFGHIJKLMNOPQ   RSTUVWXYZ[\]^_`abcdef  !"#$%&'()*+  !"#$%&'()*+ *+()'&%$#"!   !"#$%&'())*++,-./0123,-./01g2hijklm301/.-,23,-./01123 456789:;<=>?@ABCDEFGHIJKLMNOPQRS?456789:;<=>?@ABCDEFGHIJKLMNOPQRSnopqrstuvwxyz{|}~ RSPQMNOLKJIHGFEDCBA@?>=<;:987654 456789:;<=>?@ABCDEFGHIJKLMNONOPQQRSSTUVWXY1TUVWXYTUVWXYTUUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxAZ[\]^_`abcdefghijklmnopqrstuvwxvwxtursqponmlkjihgfedbca`_^]\[ZZ[\]^_`abccdefghijklmnopqrsstuuvwwx yz{|}~yz{|}~ {~}|yz yzz{~}||}~  "     !"#$%& '()*+,-./0123456789:;<  =>?@ABCDEF GHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~4'"    %   !"#$%&' ()*+,-./01234567%89:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[ !"#$%&'()*+,-./0123456789:;<=>?@ABCCDEFGHIJJKLMNOPQRSTUVWXYZ[\]^_`abcdefghiijjklmnopqrstuvwxyyz{|}~        J C              %#     # !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~+                   !                                                                          !0/"#$%&'()*+,-./0123456789:;<=>?@ABC!D EFGHIJKLMNOPQRSTUVWXYZ[\]^_`a'bcdefghi6jklmnopqrstuvwxyz{|}~r'(k*+HaskellForMaths-0.1.5"Math.Algebra.Group.StringRewriting#Math.Algebra.Group.PermutationGroupMath.Combinatorics.GraphMath.Common.IntegerAsType!Math.Algebra.Commutative.MonomialMath.Algebra.Field.BaseMath.Algebra.Commutative.MPolyMath.Algebra.Field.Extension"Math.Algebra.NonCommutative.NCPoly)Math.Algebra.NonCommutative.TensorAlgebra%Math.Projects.KnotTheory.LaurentMPolyMath.Projects.KnotTheory.Braid&Math.Projects.KnotTheory.TemperleyLieb%Math.Projects.KnotTheory.IwahoriHecke!Math.Combinatorics.FiniteGeometryMath.Combinatorics.Design'Math.Combinatorics.StronglyRegularGraphMath.Combinatorics.HypergraphMath.Projects.RootSystem(Math.Projects.ChevalleyGroup.ExceptionalMath.Common.ListSetMath.Algebra.Group.SchreierSimsMath.Combinatorics.GraphAutsMath.Algebra.Commutative.GBasis#Math.Algebra.NonCommutative.GSBasisMath.Algebra.LinearAlgebra&Math.Projects.ChevalleyGroup.ClassicalS PermutationP.^p^--^eltsGraphG nullGraphT97T89T83T79T73T71T67T61T59T53T47T43T41T37T31T29T23T19T17T13T11T7T5T3T2TOneTZeroTMinus1M IntegerAsTypevalueElimGrevlexGlexLexMonomialconvertMsupportMF97F89F83F79F73F71F67F61F59F53F47F43F41F37F31F29F23F19F17F13F11F7F5F3F2 FiniteFieldeltsFqbasisFqFpQMPolyMPtoLextoGlex toGrevlextoElim QSqrtMinus5 QSqrtMinus3 QSqrtMinus2 QSqrtMinus1QSqrt7QSqrt5QSqrt3QSqrt2SqrtF32 ConwayF32F27 ConwayF27F25 ConwayF25F16 ConwayF16F9ConwayF9F8ConwayF8F4ConwayF4ExtensionFieldExtPolynomialAsTypepvalueUPolyUP quotRemUP InvertibleinvVarZYXNPolyNPWeylGensDBasisE LaurentMPolyLPLaurentMonomialLM BraidGensLPQTemperleyLiebGensIwahoriHeckeGensT ZeroOneStarStarOneZeroDesign DesignVertexBC Incidence HypergraphHTypeFAbasisEltOctonionOrewrite rewrite''splitSubstring findOverlap knuthBendix1ordpairshortlex knuthBendix2merge knuthBendix3 knuthBendixnfss_s1s2s3_Stri_D toListSet isListSetunion intersect\\symDiffdisjointisSubsetrotateL fromPairs fromPairs'toPairsfromListsupp fromCyclestoCyclescycleOfparitysignorderEltinverse~^commclosureSclosureorbit.^^orbitPorbitV-^^orbitBorbitEactionorbits_C_D2_AdpwrtoSneltsSorderisMemberorderTGSeltsTGSgens~^^ conjClass conjClassReps centralizercentre normalizer stabilizerptStabsetStab reduceGens normalClosure commutatorGp derivedSubgpisSubgpisNormal**^cosets cosetAction quotientGp//~~^conjugateSubgps subgpAction cosetRepsGxschreierGeneratorsGxsiftfindBasebsgsbsgs'newLevel newLevel'ssss' isMemberBSGSeltsBSGScartProd orderBSGSindexsgsreduceGensBSGSsetpowersetcombinationsOf isSetSystemisGraphgraphtoGraphverticesedgesincidenceMatrixfromIncidenceMatrixadjacencyMatrixfromAdjacencyMatrixckkbkb'q'q tetrahedroncube octahedron dodecahedron icosahedronto1npetersen complement lineGraph lineGraph'sizevalency valencies regularParam isRegularisCubicnbrs findPathsdistancediameter findCyclesgirthdistancePartition component isConnectedjkneserjohnsonisVertexTransitiveisEdgeTransitive->^isArcTransitiveisArcTransitive'findArcsisnArcTransitiveis2ArcTransitiveis3ArcTransitiveisDistanceTransitiverefine isGraphAut graphAuts1 graphAuts2 graphAuts3 isSingleton graphAuts removeGens graphAutsNew graphIsosisIsodiffsdegMdividesMproperlyDividesMlcmMgcdMcoprimeM numeratorQ denominatorQextendedEuclid primitiveEltpowerscharf2f3f5f7f11f13f17f19f23f29f31f37f41f43f47f53f59f61f67f71f73f79f83f89f97cmpTerm mergeTermscollectvarabdstuvwxyzx_x0x1x2x3 convertMPvarLexvarElimltlmdegmulTdivTdividesTproperlyDividesTlcmT.* quotRemMP%%divModMPdivMPmodMPinjecttoMonictoZsubstsupportsPolyisGBgb1pairWithreducegb2pairs!gb2bgb3bcmpFstmergeBygb4bgbsugar cmpNormalcmpSuggb3gb4toUPoly<+><*>convertmonomialmodUPextendedEuclidUPembedpolysf4x4f8x8f9x9f16x16f25x25f27x27f32x32sqrt2sqrt3sqrt5sqrt7i sqrtminus2 sqrtminus3 sqrtminus5divMlc quotRemNPremNPremNP2gb'gb2'mbasisQAe_e1e2e3e4dim tensorBasis extRelationsextnf exteriorBasis symRelationssymnfsymmetricBasis weylRelationsweylnf weylBasisd_d1d2d3deltaweylRelations'weylnf' weylBasis'degLM denominatorLMlcmLMdivLM quotRemLPreduceLP denominatorLPsqrtvar^^^s4writhek3_1k4_1k5_1k7_1d' tlRelationsdimTLtlnftlBasistr'tra' fromBraidjonest_t1t2t3t4z' ihRelationsdimIHihnfihBasistau'tauhomflylmhomfly'homfly''coeffsjones'<->*>*>><.><<+>><<->><<*>><<*><*>>fMatrixfMatrix'idMxjMxzMxinverse1inverse2rowEchelonFormreducedRowEchelonFormkernel kernelRREdetptsAGptsPGpnfispnf closureAG closurePGqtorialqnomial numFlatsPG numFlatsAGqtorialsqnomialsrrefsflatsPGflatsAGdesigntoDesignisValidpointsblocksnoRepeatedBlocks tDesignParamsfindvk findlambda designParams isStructureisDesign is2DesignisSquare subsetDesign pairDesignag2pg2 flatsDesignPGpg flatsDesignAGag paleyDesign fanoPlanedual derivedDesign pointResidualcomplementaryDesign blockResidual isDesignAutincidenceGraph designAuts1 designAuts designAutsNew alphaL2_23 betaL2_23 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factorialtest1test2numPtsAGnumPtsPGslelemTransvectionorderLorderS2orderS omegaevenomegaoddoi0i1i2i3i4i5i6toListexposenfconjsqnorm isOrthogonal antiCommutes octonionsisUnitunitImagOctonionsautFrom%^alpha3beta3gamma3sgamma3alpha3'beta3'gamma3'alpha4beta4gamma4sgamma4alpha4'beta4'gamma4'