s\/      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~        !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\] ^ _!`!a!b!c!d!e!f"g"h"i"j"k"l"m"n#o#p#q#r#s#t#u#v#w#x$y$z${$|$}$~$$$$$$$$$$$%%%%%%%%%%%%&&&&&&&&&&&&&&&&&&&&&&&''''(((((((((((((((((()))))*********++++,,,,----------0xGiven a field type k (ie a Fractional instance), Vect k b is the type of the free k-vector space over the basis type b. I Elements of Vect k b consist of k-linear combinations of elements of b. The zero vector Addition of vectors "Addition of vectors (same as add) Negation of vector $Scalar multiplication (on the left) Same as smultL. Mnemonic is  multiply through (from the left) #Scalar multiplication on the right Same as smultR. Mnemonic is !multiply through (from the right) wConvert an element of Vect k b into normal form. Normal form consists in having the basis elements in ascending order, 3 with no duplicates, and all coefficients non-zero    IA type for constructing a basis for the tensor product of vector spaces. D The tensor product of Vect k a and Vect k b is Vect k (Tensor a b) EA type for constructing a basis for the direct sum of vector spaces. > The direct sum of Vect k a and Vect k b is Vect k (DSum a b) *Injection of left summand into direct sum +Injection of right summand into direct sum >The coproduct of two linear functions (with the same target). W Satisfies the universal property that f == coprodf f g . i1 and g == coprodf f g . i2 -Projection onto left summand from direct sum .Projection onto right summand from direct sum <The product of two linear functions (with the same source). S Satisfies the universal property that f == p1 . prodf f g and g == p2 . prodf f g ,The direct sum of two vector space elements (The direct sum of two linear functions. ] Satisfies the universal property that f == p1 . dsumf f g . i1 and g == p2 . dsumf f g . i2 0The tensor product of two vector space elements +The tensor product of two linear functions  !"#$%&'(rTrivial k is the field k considered as a k-vector space. In maths, we would not normally make a distinction here, _ but in the code, we need this if we want to be able to put k as one side of a tensor product. )*+^A bialgebra is an algebra which is also a coalgebra, subject to some compatibility conditions ,cAn instance declaration for Coalgebra k b is saying that the vector space Vect k b is a k-algebra. -./rCaution: If we declare an instance Algebra k b, then we are saying that the vector space Vect k b is a k-algebra. k In other words, we are saying that b is the basis for a k-algebra. So a more accurate name for this class  would have been AlgebraBasis. 012Monoid 3456 !"#$%&'()*+,-./0123456234/01,-.+)*(56&'$%"# ! !!"##$%%&''()**+,-.-./01012343456789DGiven a list of rewrite rules of the form (left,right), and a word, ^ rewrite it by repeatedly replacing any left substring in the word by the corresponding right :lImplementation of the Knuth-Bendix algorithm. Given a list of relations, return a confluent rewrite system. / The algorithm is not guaranteed to terminate. ;WGiven generators and a confluent rewrite system, return (normal forms of) all elements <FGiven generators and relations, return (normal forms of) all elements 789:;<9:;<787889:;<.=iA type for permutations, considered as functions or actions which can be performed on an underlying set. >?Wx .^ g returns the image of a vertex or point x under the action of the permutation g. 8 The dot is meant to be a mnemonic for point or vertex. @Vb -^ g returns the image of an edge or block b under the action of the permutation g. ? The dash is meant to be a mnemonic for edge or line or block. A/Construct a permutation from a list of cycles.  For example, p [[1,2,3],[4,5]]K returns the permutation that sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, 5 to 4 B'A trick: g^-1 returns the inverse of g C(g ~^ h returns the conjugate of g by h. Q The tilde is meant to a mnemonic, because conjugacy is an equivalence relation. DOx .^^ gs returns the orbit of the point or vertex x under the action of the gs EMb -^^ gs returns the orbit of the block or edge b under the action of the gs F<_C n returns generators for Cn, the cyclic group of order n G<_S n returns generators for Sn, the symmetric group on [1..n] H>_A n returns generators for An, the alternating group on [1..n] ISGiven generators for a group, return a (sorted) list of all elements of the group. U Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) JVGiven generators for a group, return the order of the group (the number of elements). U Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) KJGiven a strong generating set, return the order of the group it generates LdconjClassReps gs returns a conjugacy class representatives and sizes for the group generated by gs. E This implementation is only suitable for use with small groups (|G| < 10000). MKReturn the subgroups of a group. Only suitable for use on small groups (eg < 100 elts) NisNormal gs ks returns True if <ks> is normal in <gs>.  Note, it is caller' s responsibility to ensure that <ks> is a subgroup of <gs> (ie that each k is in <gs>). ORReturn the normal subgroups of a group. Only suitable for use on small groups (eg < 100 elts) PquotientGp gs ks returns <gs> / <ks> QSynonym for quotientGp R=>   ?@ A BCDEFGH !"#$%I&J'()*+K,-.L/0M123456789:;<=NO>?PQ@AB=>?@ABCDEFGHIJKLMNOPQ=>>?@ABCDEFGHIJKLMNOPQRJGiven generators for a permutation group, return a strong generating set. q The result is calculated using Schreier-Sims algorithm, and is relative to the base implied by the Ord instance SvGiven generators for a group, determine whether a permutation is a member of the group, using Schreier-Sims algorithm TqGiven generators for a group, return a (sorted) list of all elements of the group, using Schreier-Sims algorithm UtGiven generators for a group, return the order of the group (the number of elements), using Schreier-Sims algorithm CDEFRGHIJKLMNOPSTUQRSTUVWXRSTURSTUVWXJGiven generators for a permutation group, return a strong generating set. P The result is calculated using random Schreier-Sims algorithm, so has a small (<#10^-6) chance of being incomplete. > The sgs is relative to the base implied by the Ord instance. YTGiven a strong generating set gs, isMemberSGS gs is a membership test for the group YVWZX[\]^_`YVWXYVWXYZ[zGiven a group gs and a transitive constituent ys, return the kernel and image of the transitive constituent homomorphism. d That is, suppose that gs acts on a set xs, and ys is a subset of xs on which gs acts transitively. m Then the transitive constituent homomorphism is the restriction of the action of gs to an action on the ys. \[Given a transitive group gs, find all non-trivial block systems. That is, if gs act on xs, y find all the ways that the xs can be divided into blocks, such that the gs also have a permutation action on the blocks ]<A more efficient version of blockSystems, if we have an sgs ^HA permutation group is primitive if it has no non-trivial block systems _`nGiven a transitive group gs, and a block system for gs, return the kernel and image of the block homomorphism 8 (the homomorphism onto the action of gs on the blocks) abcdeZ[fg\]^_`hijkZ[\]^_`Z[\]^_` aLDatatype for graphs, represented as a list of vertices and a list of edges. g Both the list of vertices and the list of edges, and also the 2-element lists representing the edges, < are required to be in ascending order, without duplicates. bc9combinationsOf k xs returns the subsets of xs of size k. K If xs is in ascending order, then the returned list is in ascending order d=Safe constructor for graph from lists of vertices and edges. K graph (vs,es) checks that vs and es are valid before returning the graph. ef&c n is the cyclic graph on n vertices g(k n is the complete graph on n vertices hHGiven a graph with vertices which are lists of small integers, eg [1,2,3], f return a graph with vertices which are the numbers obtained by interpreting these as digits, eg 123. b The caller is responsible for ensuring that this makes sense (eg that the small integers are all < 10) i:Given a graph with vertices which are lists of 0s and 1s, e return a graph with vertices which are the numbers obtained by interpreting these as binary digits.  For example, [1,1,0] -> 6. jJThe diameter of a graph is maximum distance between two distinct vertices kIThe girth of a graph is the size of the smallest cycle that it contains. M Note: If the graph contains no cycles, we return -1, representing infinity. l-kneser n k returns the kneser graph KG n,k - 2 whose vertices are the k-element subsets of [1..n]&, with edges joining disjoint subsets ?ablmcnodpqrstuvefgwxyz{|}~hijkl cabdefghijkl abbcdefghijkl mnoFThe Cayley graph (undirected) on the generators (and their inverses), # for a group given as permutations pFThe Cayley graph (undirected) on the generators (and their inverses), / for a group given as generators and relations mnopmnopmnnop q^Given a graph g, graphAuts g returns a strong generating set for the automorphism group of g. qqq/! rstuvwxyz{|}~!rstuvwxyz{|}~~z{|}yuvwxtsrrstuvwxvwxyz{|}{|}~ /Phantom type for an elimination term ordering. u In the ordering, xis come before yjs come before zks, but within the xis, or yjs, or zks, grevlex ordering is used 0Phantom type representing grevlex term ordering -Phantom type representing glex term ordering ,Phantom type representing lex term ordering $2F7 is a type for the finite field with 7 elements 2F5 is a type for the finite field with 5 elements 2F3 is a type for the finite field with 3 elements 2F2 is a type for the finite field with 2 elements RQ is just the rationals, but with a better show function than the Prelude version f2 lists the elements of F2 f3 lists the elements of F3 f5 lists the elements of F5 f7 lists the elements of F7 ?     $$#Type for multivariate polynomials. f ord is a phantom type defining how terms are ordered, r is the type of the ring we are working over. q For example, a common choice will be MPoly Grevlex Q, meaning polynomials over Q with the grevlex term ordering *Create a variable with the supplied name. f By convention, variable names should usually be a single letter followed by none, one or two digits. *Convert a polynomial to lex term ordering +Convert a polynomial to glex term ordering .Convert a polynomial to grevlex term ordering 1 !"#$%&'()*+,mGiven a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials -./0123456789:;<=>? F@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcde        RCreate a non-commutative variable for use in forming non-commutative polynomials. & For example, we could define x = var x , y = var y . Then x*y /= y*x.      fghijklmnopqrstuvw              0 xyz{|}~"  !"1In effect, we have (Num k, Monomial m) => Monad (v8 -> Vect k (m v)), with return = var, and (>>=) = bind.  However, we can'Mt express this directly in Haskell, firstly because of the Ord b constraint,  secondly because Haskell doesn't support type functions. # !"#  !"#  !!"#$%&'()*+,-./01$%&'()*+,-./01+-,&*)('/01.$%$%%&*)(''()*+-,,-./01232323234564565644566789:;< 789:;<;<9:787889::;<<=>?@ABCD=>?@ABCDCD@AB?=>=>>?@ABABCDD EFGHIJKLM EFGHIJKLM FJIHGELMK EFJIHGGHIJKLMNOPQRSNOPQRSRSPQNONOOPQQRSSTUVWTUVWVWTUTUUVWWXYZ XYZZXYXYYZ[\[\[\[\\ ]^]^]^]^^!_`abcde*_`abcde      ced`ba__`baabcedde"fghijklmfghijklm!"#$%&'()*+,-./lmjkhifgfgghiijkklmm# nopqrstuvwnopqr01234567stuv89w:; roqpnstuvw noqppqrstuvw$xu <+>" v returns the sum u+v of vectors yu <->) v returns the difference u-v of vectors zk *>< v returns the product k*v of the scalar k and the vector v {u <.>L v returns the dot product of vectors (also called inner or scalar product) |u <*>O v returns the tensor product of vectors (also called outer or matrix product) }a <<+>># b returns the sum a+b of matrices ~a <<->>* b returns the difference a-b of matrices a <<*>>' b returns the product a*b of matrices k *>< m returns the product k*m of the scalar k and the matrix m m <<*>9 v is multiplication of a vector by a matrix on the left v <*>>: m is multiplication of a vector by a matrix on the right !iMx n is the n*n identity matrix "jMx n is the n*n matrix of all 1s "zMx n is the n*n matrix of all 0s 5The inverse of a matrix (over a field), if it exists +The determinant of a matrix (over a field) xyz{|}~<=>?@ABCDExyz{|}~xyz{|}~% _ptsAG n fq returns the points of the affine geometry AG(n,Fq), where fq are the elements of Fq cptsPG n fq returns the points of the projective geometry PG(n,Fq), where fq are the elements of Fq PflatsPG n fq k returns the k-flats in PG(n,Fq), where fq are the elements of Fq PflatsAG n fq k returns the k-flats in AG(n,Fq), where fq are the elements of Fq  The lines (1-flats) in PG(n,fq)  The lines (1-flats) in AG(n,fq) [Incidence graph of PG(n,fq), considered as an incidence structure between points and lines [Incidence graph of AG(n,fq), considered as an incidence structure between points and lines FGHIJKLMNOPQRSTUVWX  &UThe incidence matrix of a design, with rows indexed by blocks and columns by points. S (Note that in the literature, the opposite convention is sometimes used instead.) 0The affine plane AG(2,Fq), a 2-(q^2,q,1) design AThe projective plane PG(2,Fq), a square 2-(q^2+q+1,q+1,1) design The dual of a design  The incidence graph of a design DFind a strong generating set for the automorphism group of a design OGenerators for the Mathieu group M24, a finite simple group of order 244823040 \A strong generating set for the Mathieu group M24, a finite simple group of order 244823040 [A strong generating set for the Mathieu group M23, a finite simple group of order 10200960 YA strong generating set for the Mathieu group M22, a finite simple group of order 443520 OThe Steiner system S(5,8,24), with 759 blocks, whose automorphism group is M24 OThe Steiner system S(4,7,23), with 253 blocks, whose automorphism group is M23 NThe Steiner system S(3,6,22), with 77 blocks, whose automorphism group is M22 OThe Steiner system S(5,6,12), with 132 blocks, whose automorphism group is M12 NThe Steiner system S(4,5,11), with 66 blocks, whose automorphism group is M11 KGenerators for the Mathieu group M12, a finite simple group of order 95040 XA strong generating set for the Mathieu group M12, a finite simple group of order 95040 WA strong generating set for the Mathieu group M11, a finite simple group of order 7920 ?YZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~''(JIs this hypergraph uniform - meaning that all blocks are of the same size [Is this hypergraph a projective plane - meaning that any two lines meet in a unique point, ) and any two points lie on a unique line 7Is this hypergraph a projective plane with a triangle. } This is a weak non-degeneracy condition, which eliminates all points on the same line, or all lines through the same point. 9Is this hypergraph a projective plane with a quadrangle. . This is a stronger non-degeneracy condition. 1Is this hypergraph a (projective) configuration. ;The Heawood graph is the incidence graph of the Fano plane 6The Tutte-Coxeter graph, also called the Tutte 8-cage )&Are the two latin squares orthogonal? HAre the latin squares mutually orthogonal (ie each pair is orthogonal)? MOLS from a projective plane * #  +_The special linear group SL(n,Fq), generated by elementary transvections, returned as matrices DThe projective special linear group PSL(n,Fq) == A(n,Fq) == SL(n,Fq)/Z, 7 returned as permutations of the points of PG(n-1,Fq). 4 This is a finite simple group provided n>2 or q>3. 5The symplectic group Sp(2n,Fq), returned as matrices AThe projective symplectic group PSp(2n,Fq) == Cn(Fq) == Sp(2n,Fq)/Z, 8 returned as permutations of the points of PG(2n-1,Fq). > This is a finite simple group for n>1, except for PSp(4,F2). ,7     - >Generators for G2(3), a finite simple group of order 4245696, D as a permutation group on the 702 unit imaginary octonions over F3 & !"#$%&'()*+,-.  /123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxymz{|}~mz                                                   !"#$%&'()*+,-./01234567895:2;<=>?@ABCCD9EFGH45IJKLMNOPQRS;TU>VWXYZ[\]^_`)abcdefLMgihi2 j k!l!m!n!o!p!q!r"s"s"t"t"u"u"v"v#w#p#q#r#x#y#z#{#|#}$7$~$:$$$$$$$$$$$$$$%%%]%%%%%%%%%&&9&&&&&&&&&&&&&&&&&&&&&''F'o'E(((((((((((((((((()))))****2*9*E*F*G*++++,,,","----A-B-----x........  T    !"#$%&'()*+,-./0123456789:;~<=#$%>?@ABCDEFGHI!JKL#MN O P Q R S T U  V W X Y Z [ \ ] ^ _ ` a b c d e f z g h i j k l m n o p q r s t _ u v w x y z { | } ~                                   /////////////////////////////////              j>B     7 !"#$%&'()*+,-./01^234T567B8s000000900:0;<=>?@ABCDEFGHIJKL   MNOPQRS\[MTUVAW567BXYZ[5\]^_`abcdef<g=hijklmnop q r s t u \  [ v w x y z { | o } ^  ~    !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"""""""""""""""############$$$$$$ $$$$s%%%%%%%%%%%%%%%%%%%&&&&&&&&&&&&&&&&&&&&&&b&&&&&&&&&&&&&&&&&&''''''''''''''''''''''''' ' '' ' ' '''''(((((((V(((((())) )!)")#)$)%)&*'***(*)***+*,*-*.*/*0*1*2*3*4*5*6*7*8*9*:*;*<*=*>+?+@+A+B++C+D+E++F++G,,",,H,I,J,^,_,,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,[,\,],^,_,`,a,b,c,d,e,f,<,g,h,i,z,,j,k,l,m,n,o,p,q--r-s-=-~-H-t--u-v-w-x-y-z-{-|-}-~----------HaskellForMaths-0.3.2Math.Algebras.VectorSpaceMath.Algebras.TensorProductMath.Algebras.Structures"Math.Algebra.Group.StringRewriting#Math.Algebra.Group.PermutationGroupMath.Algebra.Group.SchreierSims%Math.Algebra.Group.RandomSchreierSimsMath.Algebra.Group.SubquotientsMath.Combinatorics.GraphMath.Algebra.Group.CayleyGraphMath.Combinatorics.GraphAuts"Math.QuantumAlgebra.TensorCategoryMath.Common.IntegerAsType!Math.Algebra.Commutative.MonomialMath.Algebra.Field.BaseMath.Algebra.Commutative.MPolyMath.Algebra.Commutative.GBasisMath.Algebra.Field.Extension"Math.Algebra.NonCommutative.NCPoly)Math.Algebra.NonCommutative.TensorAlgebraMath.Algebras.CommutativeMath.Algebras.AffinePlaneMath.Algebras.GroupAlgebraMath.Algebras.LaurentPolyMath.Algebras.MatrixMath.Algebras.NonCommutativeMath.Algebras.QuaternionsMath.Algebras.TensorAlgebra%Math.Projects.KnotTheory.LaurentMPolyMath.Projects.KnotTheory.Braid&Math.Projects.KnotTheory.TemperleyLieb%Math.Projects.KnotTheory.IwahoriHecke"Math.QuantumAlgebra.OrientedTangle Math.QuantumAlgebra.QuantumPlaneMath.QuantumAlgebra.TangleMath.Algebra.LinearAlgebra!Math.Combinatorics.FiniteGeometryMath.Combinatorics.Design'Math.Combinatorics.StronglyRegularGraphMath.Combinatorics.HypergraphMath.Combinatorics.LatinSquaresMath.Projects.RootSystem&Math.Projects.ChevalleyGroup.Classical$Math.Projects.MiniquaternionGeometry(Math.Projects.ChevalleyGroup.ExceptionalMath.Common.ListSetMath.Projects.Rubik#Math.Algebra.NonCommutative.GSBasisEBasisEVectVzeroadd<+>negsmultL*>smultR<*nflinearTensorDSumi1i2coprodfp1p2prodfdsumedsumftetfassocLassocRdistrLundistrLdistrRundistrRComodulecoactionModuleactionMonoidCoalgebraMC SetCoalgebraSCTrivial HopfAlgebraantipode Bialgebra CoalgebracounitcomultAlgebraunitmultMonmunitmmultunit'counit'SGenSrewrite knuthBendixnfselts PermutationP.^-^p^-~^.^^-^^_C_S_AorderorderSGS conjClassRepssubgpsisNormal normalSubgps quotientGp//sgsisMember initProdRepl nextProdRepl isMemberSGS isTransitive!transitiveConstituentHomomorphism blockSystemsblockSystemsSGS isPrimitiveisPrimitiveSGSblockHomomorphismGraphGcombinationsOfgraph nullGraphck fromDigits fromBinarydiametergirthkneserDigraphDG cayleyGraphP cayleyGraphS graphAutsCob2BraidSymmetricGroupoidWeakTensorCategoryassoclunitrunitStrictTensorCategoryTensorCategorytunittobtarCategoryObArid_sourcetarget>>>T97T89T83T79T73T71T67T61T59T53T47T43T41T37T31T29T23T19T17T13T11T7T5T3T2TOneTZeroTMinus1M IntegerAsTypevalueElimGrevlexGlexLexMonomialconvertMsupportMF97F89F83F79F73F71F67F61F59F53F47F43F41F37F31F29F23F19F17F13F11F7F5F3F2 FiniteFieldeltsFqbasisFqFpQf2f3f5f7MPolyMPvarabdstuvwxyzx0x1x2x3toLextoGlex toGrevlextoElimgb QSqrtMinus5 QSqrtMinus3 QSqrtMinus2 QSqrtMinus1QSqrt7QSqrt5QSqrt3QSqrt2SqrtF32 ConwayF32F27 ConwayF27F25 ConwayF25F16 ConwayF16F9ConwayF9F8ConwayF8F4ConwayF4ExtensionFieldExtPolynomialAsTypepvalueUPolyUP quotRemUP conjugate InvertibleinvVarZYXNPolyNPWeylGensDBasis DivisionBasisdividesBdivBpowersGlexPoly GlexMonomialbind%%SL2ABCDCBAXY GroupAlgebraip LaurentPolyLaurentMonomialLMM3E3Mat2'E2'Mat2E2divMNCPolyNonComMonomialNCM QuaternionHBasisKJIOneijExteriorAlgebraSymmetricAlgebraSym TensorAlgebraTA LaurentMPolyLP BraidGensLPQTemperleyLiebGensIwahoriHeckeGensTOrientedTangleHorizDirToRToLOrientedMinusPlusSL2qM2qAq02Aq20 TangleRepTanglecapcupoverunderkauffman<-><.><*><<+>><<->><<*>>*>><<*><*>>iMxjMxzMxinversereducedRowEchelonFormdet ZeroOneStarStarZeroptsAGptsPGflatsPGflatsAGlinesPGlinesAGincidenceGraphPGincidenceGraphAGDesignincidenceMatrixag2pg2dual derivedDesign pointResidual blockResidualincidenceGraph designAutsm24m24sgsm23sgsm22sgss_5_8_24s_4_7_23s_3_6_22s_5_6_12s_4_5_11m12m12sgsm11sgs DesignVertex HypergraphH isUniformisPartialLinearSpaceisProjectivePlaneisProjectivePlaneTriisProjectivePlaneQuadisGeneralizedQuadrangleisConfiguration fanoPlane heawoodGraphdesarguesConfigurationdesarguesGraphpappusConfiguration pappusGraph coxeterGraphtutteCoxeterGraph findLatinSqs isLatinSq isOrthogonalisMOLSfromProjectivePlaneTypeFbasisEltsllsp2s2J9OctonionOi0i3i4i5i6g2_3addmergeee1e2e3unitInLunitOutLunitInRunitOutRtwist*.rewrite1splitSubstring findOverlap knuthBendix1ordpairshortlex knuthBendix2merge knuthBendix3s_s1s3_S'tri_D toListSet isListSetunion intersect\\symDiffdisjointisSubsetrotateL fromPairs fromPairs'toPairsfromListsupp fromCyclestoCyclescycleOfparitysignorderEltcommclosureSclosureorbitorbitPorbitVorbitBorbitEorbits_D2dpwrtoSn fromDigits' fromBinary'eltsSminsupporderTGSeltsTGS tgsFromSgsgens~^^ conjClass reduceGensisSubgp isMinimal centralizercentre normalizer stabilizerptStabsetStab normalClosure commutatorGp derivedSubgp-*--**-isSimplecosets~~^conjugateSubgps subgpAction cosetRepsGxschreierGeneratorsGxsiftfindBasebsgsbsgs'newLevel newLevel'ssss' isMemberBSGSeltsBSGScartProd orderBSGSindexreduceGensBSGS testProdRepl updateArrayrssrss' initLevels updateLevels updateLevels'baseTransversalsSGSisLeftisRightunRight restrictLeft"transitiveConstituentHomomorphism' minimalBlockblockHomomorphism'intersectionNormalClosurecentralizerSymTranssetpowerset isSetSystemisGraphtoGraphverticesedgesfromIncidenceMatrixadjacencyMatrixfromAdjacencyMatrixkbkb'q'q tetrahedroncube octahedron dodecahedron icosahedronto1npetersen complement lineGraph lineGraph'sizevalency valenciesvalencyPartition regularParam isRegularisCubicnbrs findPathsdistance findCyclesdistancePartition component isConnectedjohnsonbipartiteKneser desargues1gp petersen2prismdurer mobiusKantor dodecahedron2 desargues2toSetcayleyDigraphPcayleyDigraphSfromTranspositions fromTrans bubblesorttoTranstoTranspositions inversionsisVertexTransitiveisEdgeTransitive->^isArcTransitiveisArcTransitive'findArcsisnArcTransitiveis2ArcTransitiveis3ArcTransitiveisDistanceTransitiverefinerefine' isGraphAut graphAuts1 graphAuts2 graphAuts3 isSingleton graphAuts4eqgraph toEquitable toEquitable2 splitNumNbrs dfsEquitable incidenceAuts graphIsosisIsofr rubikCube cornerFaces edgeFaceskerCornerFaces imCornerFaces 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