  
  [1X6 [33X[0;0YDatabases of Residue-Class-Wise Affine Groups and -Mappings[133X[101X
  
  [33X[0;0YThe  [5XRCWA[105X  package  contains  a  number of databases of rcwa groups and rcwa
  mappings.  They  can be loaded into a [5XGAP[105X session by the functions described
  in this chapter.[133X
  
  
  [1X6.1 [33X[0;0YThe collection of examples[133X[101X
  
  [1X6.1-1 LoadRCWAExamples[101X
  
  [29X[2XLoadRCWAExamples[102X(  ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  record  containing  a collection of examples of rcwa groups and
            -mappings, as stored in the file [11Xpkg/rcwa/examples/examples.g[111X.[133X
  
  [33X[0;0YThe  components  of  the  record returned by this function are records which
  contain  the  individual groups and mappings. A detailed description of some
  of the examples can be found in Chapter [14X7[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xexamples := LoadRCWAExamples();;                                 [127X[104X
    [4X[25Xgap>[125X [27XRecNames(examples);[127X[104X
    [4X[28X[ "Basics", "CollatzMapping", "HigmanThompson", "CTPZ", "CT3Z", [128X[104X
    [4X[28X  "OddNumberOfGens_FiniteOrder", "ZxZ", "SlowlyContractingMappings", [128X[104X
    [4X[28X  "MatthewsLeigh", "HicksMullenYucasZavislak", "CollatzlikePerms", [128X[104X
    [4X[28X  "GF2xFiniteCycles", "GrigorchukQuotients", "F2_PSL2Z", [128X[104X
    [4X[28X  "MaybeInfinitelyPresentedGroup", "Hexagon", "FiniteQuotients", [128X[104X
    [4X[28X  "ClassTranspositionProducts", "Venturini", "Farkas", [128X[104X
    [4X[28X  "SymmetrizingCollatzTree", "FiniteVsDenseCycles", [128X[104X
    [4X[28X  "AbelianGroupOverPolynomialRing", "Semilocals", [128X[104X
    [4X[28X  "LongCyclesOfPrimeLength", "ModuliOfPowers", [128X[104X
    [4X[28X  "ClassTranspositionsAsCommutators", "CoprimeMultDiv", [128X[104X
    [4X[28X  "TameGroupByCommsOfWildPerms", "CheckingForSolvability", "Syl3_S9", [128X[104X
    [4X[28X  "ClassSwitches", "CollatzFactorizationOld" ][128X[104X
    [4X[25Xgap>[125X [27XAssignGlobals(examples.ZxZ);[127X[104X
    [4X[28XThe following global variables have been assigned:[128X[104X
    [4X[28X[ "R", "b", "a", "twice", "twice1", "twice2", "switch", "reflection", [128X[104X
    [4X[28X  "reflection1", "reflection2", "transvection", "hyperbolic", "T2", [128X[104X
    [4X[28X  "Sigma_T", "SigmaT", "SigmaTm", "commT_Tm" ][128X[104X
    [4X[25Xgap>[125X [27Xa*b = Sigma_T;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDisplay(Sigma_T);[127X[104X
    [4X[28X[128X[104X
    [4X[28XRcwa mapping of Z^2 with modulus (1,0)Z+(0,6)Z[128X[104X
    [4X[28X[128X[104X
    [4X[28X            /[128X[104X
    [4X[28X            | (2m+1,(3n+1)/2) if (m,n) in (0,1)+(1,0)Z+(0,2)Z[128X[104X
    [4X[28X            | (m,n/2)         if (m,n) in (0,0)+(1,0)Z+(0,6)Z U [128X[104X
    [4X[28X (m,n) |-> <                              (0,2)+(1,0)Z+(0,6)Z[128X[104X
    [4X[28X            | (2m,n/2)        if (m,n) in (0,4)+(1,0)Z+(0,6)Z[128X[104X
    [4X[28X            |[128X[104X
    [4X[28X            \[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YDatabases of rcwa groups[133X[101X
  
  [1X6.2-1 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions[101X
  
  [29X[2XLoadDatabaseOfGroupsGeneratedBy3ClassTranspositions[102X(  ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  record containing a database of all groups generated by 3 class
            transpositions which interchange residue classes with moduli [22X≤ 6[122X.[133X
  
  [33X[0;0YThe  record  presently has the components [10Xgrps[110X (the list of the 52394 groups
  --  21948 finite and 30446 infinite ones), [10Xsizes[110X (the list of group orders),
  [10Xmods[110X  (the  list  of  moduli  of the groups), [10Xtrsstatus[110X (lists what is known
  about whether the groups are transitive on the nonnegative integers in their
  support),  [10Xcts[110X  (the  list  of all 69 class transpositions which interchange
  residue  classes  with  moduli  [22X≤  6[122X),  and  possibly  further which are not
  described  here.  For  all  integers  [10Xi[110X  from  1  to  52394  it  holds  that
  [10XSize(grps[i])  =  sizes[i][110X  and  that [10XModulus(grps[i]) = mods[i][110X. Similarly,
  [10Xtrsstatus[i][110X  describes  what  is known about whether the group [10Xgrps[i][110X acts
  transitively  on  the set of nonnegative integers in its support -- for many
  of the groups this is a description of how the computation failed.[133X
  
  [33X[0;0YThe  group [10Xgrps[44132][110X might be called the [21XCollatz group[121X or the [21X[22X3n+1[122X - group[121X
  --  its  action on the set of positive integers which are not divisible by 6
  is transitive if and only if the [22X3n+1[122X conjecture is true.[133X
  
  [33X[0;0YNote that the contents of this database are not [21Xset in stone[121X, and are likely
  to change in coming releases. Also note that the database presently contains
  an entry for every unordered triple of distinct class transpositions in [10Xcts[110X,
  which means that it contains multiple copies of equal groups. For the future
  it is planned to include information on which groups are equal and which are
  isomorphic,  but in particular for the infinite groups this task seems to be
  algorithmically hard.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xdata := LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions();;[127X[104X
    [4X[25Xgap>[125X [27XViewString(data.grps[44132]); # the "3n+1 group"[127X[104X
    [4X[28X"<(2(3),4(6)),(1(3),2(6)),(1(2),4(6))>"[128X[104X
    [4X[25Xgap>[125X [27Xdata.trsstatus[44132]; # deciding this would solve the 3n+1 problem[127X[104X
    [4X[28X"exceeded memory bound"[128X[104X
    [4X[25Xgap>[125X [27XLength(Set(data.sizes));[127X[104X
    [4X[28X1066[128X[104X
    [4X[25Xgap>[125X [27XMaximum(Filtered(data.sizes,IsInt));[127X[104X
    [4X[28X7165033589793852697531456980706732548435609645091822296777976465116824959\[128X[104X
    [4X[28X2135499174617837911754921014138184155204934961004073853323458315539461543\[128X[104X
    [4X[28X4480515260818409913846161473536000000000000000000000000000000000000000000\[128X[104X
    [4X[28X000000[128X[104X
    [4X[25Xgap>[125X [27XPositions(data.sizes,last);[127X[104X
    [4X[28X[ 33814, 36548 ][128X[104X
    [4X[25Xgap>[125X [27XList(data.grps{last},ViewString);[127X[104X
    [4X[28X[ "<(1(5),4(5)),(0(3),1(6)),(3(4),0(6))>", [128X[104X
    [4X[28X  "<(0(5),3(5)),(2(3),4(6)),(0(4),5(6))>" ][128X[104X
    [4X[25Xgap>[125X [27XCollected(data.mods);[127X[104X
    [4X[28X[ [ 0, 30446 ], [ 3, 1 ], [ 4, 37 ], [ 5, 120 ], [ 6, 1450 ], [ 8, 18 ], [128X[104X
    [4X[28X  [ 10, 45 ], [ 12, 3143 ], [ 15, 165 ], [ 18, 484 ], [ 20, 528 ], [128X[104X
    [4X[28X  [ 24, 1339 ], [ 30, 2751 ], [ 36, 2064 ], [ 40, 26 ], [ 48, 515 ], [128X[104X
    [4X[28X  [ 60, 2322 ], [ 72, 2054 ], [ 80, 44 ], [ 90, 108 ], [ 96, 108 ], [128X[104X
    [4X[28X  [ 108, 114 ], [ 120, 782 ], [ 144, 310 ], [ 160, 26 ], [ 180, 206 ], [128X[104X
    [4X[28X  [ 192, 6 ], [ 216, 72 ], [ 240, 304 ], [ 270, 228 ], [ 288, 14 ], [128X[104X
    [4X[28X  [ 360, 84 ], [ 432, 36 ], [ 480, 218 ], [ 540, 18 ], [ 720, 120 ], [128X[104X
    [4X[28X  [ 810, 112 ], [ 864, 8 ], [ 960, 94 ], [ 1080, 488 ], [ 1620, 44 ], [128X[104X
    [4X[28X  [ 1920, 38 ], [ 2160, 506 ], [ 3240, 34 ], [ 3840, 12 ], [128X[104X
    [4X[28X  [ 4320, 218 ], [ 4860, 16 ], [ 6480, 282 ], [ 7680, 10 ], [128X[104X
    [4X[28X  [ 8640, 16 ], [ 12960, 120 ], [ 14580, 2 ], [ 25920, 34 ], [128X[104X
    [4X[28X  [ 30720, 2 ], [ 38880, 12 ], [ 51840, 8 ], [ 77760, 32 ] ][128X[104X
    [4X[25Xgap>[125X [27XCollected(data.trsstatus);[127X[104X
    [4X[28X[ [ "> 1 orbit (mod m)", 593 ], [128X[104X
    [4X[28X  [ "Mod(U DecreasingOn) exceeded <maxmod>", 23 ], [128X[104X
    [4X[28X  [ "U DecreasingOn stable and exceeded memory bound", 35 ], [128X[104X
    [4X[28X  [ "U DecreasingOn stable for <maxeq> steps", 5859 ], [128X[104X
    [4X[28X  [ "exceeded memory bound", 599 ], [ "finite", 21948 ], [128X[104X
    [4X[28X  [ "intransitive, but finitely many orbits", 8 ], [128X[104X
    [4X[28X  [ "seemingly finite and infinite orbits (exponential growth)", 2981 ], [128X[104X
    [4X[28X  [ "seemingly finite and infinite orbits (growth unclear)", 86 ], [128X[104X
    [4X[28X  [ "seemingly finite and infinite orbits (linear growth)", 10877 ], [128X[104X
    [4X[28X  [ "seemingly only finite orbits (long)", 778 ], [128X[104X
    [4X[28X  [ "seemingly only finite orbits (medium)", 1395 ], [128X[104X
    [4X[28X  [ "seemingly only finite orbits (short)", 4603 ], [128X[104X
    [4X[28X  [ "seemingly only finite orbits (very long)", 1510 ], [128X[104X
    [4X[28X  [ "seemingly only finite orbits (very long, very unclear)", 4 ], [128X[104X
    [4X[28X  [ "seemingly only finite orbits (very short)", 765 ], [128X[104X
    [4X[28X  [ "transitive", 330 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.2-2 LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions[101X
  
  [29X[2XLoadDatabaseOfGroupsGeneratedBy4ClassTranspositions[102X(  ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  record containing a database of all groups generated by 4 class
            transpositions  which  interchange residue classes with moduli [22X≤ 6[122X
            for which all subgroups generated by 3 out of the 4 generators are
            finite.[133X
  
  [33X[0;0YThe  record  presently  has  the  components  [10Xgrps4_3finite[110X (the list of all
  140947  groups  in  the  database), [10Xsizes4[110X (the list of group orders), [10Xmods4[110X
  (the list of moduli of the groups), [10Xconjugacyclasses4cts[110X (a list of lists of
  positions  of groups in the list [10Xgrps4_3finite[110X which are already known to be
  conjugate),  [10Xgrps4_3finite_reps[110X  (tentative  conjugacy class representatives
  from  the  list  [10Xgrps4_3finite[110X -- [13Xtentative[113X in the sense that likely some of
  the  groups  in the list are still conjugate), [10Xcts[110X (the list of all 69 class
  transpositions   which   interchange  residue  classes  with  moduli  [22X≤  6[122X),
  [10Xgrps4_3finitepos[110X  (the  list  obtained from [10Xgrps4_3finite[110X by replacing every
  group  generator by its position in the list [10Xcts[110X, and possibly further which
  are  not  described  here. For all integers [10Xi[110X from 1 to 140947 it holds that
  [10XSize(grps4_3finite[i])  =  sizes4[i][110X  and  that  [10XModulus(grps4_3finite[i]) =
  mods4[i][110X.  Note that the contents of this database are not [21Xset in stone[121X, and
  are  likely  to  change  in  coming  releases.  Also  note that the database
  contains  an  entry  for  every suitable unordered 4-tuple of distinct class
  transpositions in [10Xcts[110X, which means that it contains multiple copies of equal
  groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xdata := LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions();;[127X[104X
    [4X[25Xgap>[125X [27XAssignGlobals(data);[127X[104X
    [4X[28XThe following global variables have been assigned:[128X[104X
    [4X[28X[ "cts", "grps4_3finitepos", "grps4_3finite", "conjugacyclasses4cts", [128X[104X
    [4X[28X  "grps4_3finite_reps", "mods4", "sizes4set", "sizes4pos", "sizes4" ][128X[104X
    [4X[25Xgap>[125X [27XLength(grps4_3finite);[127X[104X
    [4X[28X140947[128X[104X
    [4X[25Xgap>[125X [27XLength(sizes4);[127X[104X
    [4X[28X140947[128X[104X
    [4X[25Xgap>[125X [27XSize(grps4_3finite[1]);[127X[104X
    [4X[28X518400[128X[104X
    [4X[25Xgap>[125X [27Xsizes4[1];[127X[104X
    [4X[28X518400[128X[104X
    [4X[25Xgap>[125X [27XMaximum(Filtered(sizes4,IsInt));[127X[104X
    [4X[28X<integer 420...000 (3852 digits)>[128X[104X
    [4X[25Xgap>[125X [27XModulus(grps4_3finite[1]);[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27Xmods4[1];[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XLength(Set(sizes4));[127X[104X
    [4X[28X7339[128X[104X
    [4X[25Xgap>[125X [27XLength(Set(mods4));[127X[104X
    [4X[28X91[128X[104X
    [4X[25Xgap>[125X [27XSet(mods4);[127X[104X
    [4X[28X[ 0, 4, 5, 6, 8, 10, 12, 15, 18, 20, 24, 30, 36, 40, 48, 60, 72, 80, 90, [128X[104X
    [4X[28X  96, 108, 120, 144, 160, 180, 192, 216, 240, 270, 288, 320, 360, 384, [128X[104X
    [4X[28X  432, 480, 540, 576, 720, 810, 864, 960, 1080, 1440, 1620, 1728, 1920, [128X[104X
    [4X[28X  2160, 2430, 2592, 2880, 3240, 3840, 4320, 4860, 5760, 6480, 7680, [128X[104X
    [4X[28X  8640, 9720, 10368, 12960, 14580, 15360, 17280, 19440, 25920, 30720, [128X[104X
    [4X[28X  34560, 38880, 43740, 51840, 61440, 69120, 77760, 103680, 116640, [128X[104X
    [4X[28X  122880, 155520, 207360, 233280, 311040, 349920, 414720, 466560, [128X[104X
    [4X[28X  622080, 933120, 1244160, 1658880, 1866240, 5598720, 33592320 ][128X[104X
    [4X[25Xgap>[125X [27Xconjugacyclasses4cts{[1..4]};[127X[104X
    [4X[28X[ [ 1, 23, 563, 867 ], [ 2, 859 ], [ 3, 622 ], [ 4, 16, 868, 873 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgrps4_3finite[1] = grps4_3finite[23];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgrps4_3finite[4] = grps4_3finite[16];[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XSize(grps4_3finite[4]);[127X[104X
    [4X[28X44696597299200000[128X[104X
    [4X[25Xgap>[125X [27XSize(grps4_3finite[16]);[127X[104X
    [4X[28X44696597299200000[128X[104X
    [4X[25Xgap>[125X [27XRepresentativeAction(RCWA(Integers),grps4_3finite[4],[127X[104X
    [4X[25X>[125X [27X                                       grps4_3finite[16],OnPoints);[127X[104X
    [4X[28X( 0(30), 6(30), 12(30) ) ( 1(30), 7(30), 13(30) ) ( 2(30), 8(30), 14(30) \[128X[104X
    [4X[28X) ( 3(30), 9(30), 15(30) ) ( 4(30), 10(30), 16(30) ) ( 5(30), 11(30), 17(\[128X[104X
    [4X[28X30) )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YDatabases of rcwa mappings[133X[101X
  
  [1X6.3-1 LoadDatabaseOfProductsOf2ClassTranspositions[101X
  
  [29X[2XLoadDatabaseOfProductsOf2ClassTranspositions[102X(  ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  record  containing  a  database  of  all  products  of  2 class
            transpositions which interchange residue classes with moduli [22X≤ 6[122X.[133X
  
  [33X[0;0YThere  are  69  class  transpositions which interchange residue classes with
  moduli  [22X≤  6[122X, thus there is a total of [22X(69 ⋅ 68)/2 = 2346[122X unordered pairs of
  distinct  such  class  transpositions.  Looking  at intersection- and subset
  relations  between  the  4  involved  residue classes, we can distinguish 17
  different [21Xintersection types[121X (or 18, together with the trivial case of equal
  class  transpositions).  The  intersection type does not fully determine the
  cycle structure of the product. -- In total, we can distinguish 88 different
  cycle  types of products of 2 class transpositions which interchange residue
  classes with moduli [22X≤ 6[122X.[133X
  
  [33X[0;0YThe  components  of the returned record are a list [10XCTPairs[110X of all 2346 pairs
  of  distinct  class  transpositions  which  interchange residue classes with
  moduli  [22X≤  6[122X, functions [10XCTPairsIntersectionTypes[110X, [10XCTPairIntersectionType[110X and
  [10XCTPairProductType[110X  as  well  as  data lists [10XCTPairsProductClassification[110X and
  [10XCTPairsProductType[110X.   --   For   a   precise   description   see   the  file
  [11Xpkg/rcwa/data/ctprodclass.g[111X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xdata := LoadDatabaseOfProductsOf2ClassTranspositions();;[127X[104X
    [4X[25Xgap>[125X [27XRecNames(data);[127X[104X
    [4X[28X[ "CTPairsProductType", "CTPairs", "CTPairsIntersectionTypes", [128X[104X
    [4X[28X  "CTPairIntersectionType", "CTPairProductType", [128X[104X
    [4X[28X  "CTPairsProductClassification", "OrdersMatrix", "CTProds12", [128X[104X
    [4X[28X  "CTProds32" ][128X[104X
    [4X[25Xgap>[125X [27XLength(data.CTPairs);[127X[104X
    [4X[28X2346[128X[104X
    [4X[25Xgap>[125X [27XCollected(List(data.CTPairsProductType,l->l[2])); # order statistics[127X[104X
    [4X[28X[ [ 2, 165 ], [ 3, 255 ], [ 4, 173 ], [ 6, 693 ], [ 10, 2 ], [128X[104X
    [4X[28X  [ 12, 345 ], [ 15, 4 ], [ 20, 10 ], [ 30, 120 ], [ 60, 44 ], [128X[104X
    [4X[28X  [ infinity, 535 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.3-2 LoadDatabaseOfNonbalancedProductsOfClassTranspositions[101X
  
  [29X[2XLoadDatabaseOfNonbalancedProductsOfClassTranspositions[102X(  ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record containing a database of products of class transpositions
            which are not balanced.[133X
  
  [33X[0;0YThis  database contains a list of the 24 pairs of class transpositions which
  interchange  residue  classes  with  moduli  [22X≤  6[122X  and  whose product is not
  balanced,  as  well as a list of all 36 essentially distinct triples of such
  class transpositions whose product has coprime multiplier and divisor.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xdata := LoadDatabaseOfNonbalancedProductsOfClassTranspositions();;[127X[104X
    [4X[25Xgap>[125X [27XRecNames(data);[127X[104X
    [4X[28X[ "PairsOfCTsWhoseProductIsNotBalanced", [128X[104X
    [4X[28X  "TriplesOfCTsWhoseProductHasCoprimeMultiplierAndDivisor" ][128X[104X
    [4X[25Xgap>[125X [27XList(data.PairsOfCTsWhoseProductIsNotBalanced,[127X[104X
    [4X[25X>[125X [27X        p->List(p,TransposedClasses));[127X[104X
    [4X[28X[ [ [ 1(2), 2(4) ], [ 2(4), 3(6) ] ], [ [ 1(2), 2(4) ], [ 2(4), 5(6) ] ],[128X[104X
    [4X[28X  [ [ 1(2), 2(4) ], [ 2(4), 1(6) ] ], [ [ 1(2), 0(4) ], [ 0(4), 1(6) ] ],[128X[104X
    [4X[28X  [ [ 1(2), 0(4) ], [ 0(4), 3(6) ] ], [ [ 1(2), 0(4) ], [ 0(4), 5(6) ] ],[128X[104X
    [4X[28X  [ [ 0(2), 1(4) ], [ 1(4), 2(6) ] ], [ [ 0(2), 1(4) ], [ 1(4), 4(6) ] ],[128X[104X
    [4X[28X  [ [ 0(2), 1(4) ], [ 1(4), 0(6) ] ], [ [ 0(2), 3(4) ], [ 3(4), 4(6) ] ],[128X[104X
    [4X[28X  [ [ 0(2), 3(4) ], [ 3(4), 2(6) ] ], [ [ 0(2), 3(4) ], [ 3(4), 0(6) ] ],[128X[104X
    [4X[28X  [ [ 1(2), 2(6) ], [ 3(4), 2(6) ] ], [ [ 1(2), 2(6) ], [ 1(4), 2(6) ] ],[128X[104X
    [4X[28X  [ [ 1(2), 4(6) ], [ 3(4), 4(6) ] ], [ [ 1(2), 4(6) ], [ 1(4), 4(6) ] ],[128X[104X
    [4X[28X  [ [ 1(2), 0(6) ], [ 1(4), 0(6) ] ], [ [ 1(2), 0(6) ], [ 3(4), 0(6) ] ],[128X[104X
    [4X[28X  [ [ 0(2), 1(6) ], [ 2(4), 1(6) ] ], [ [ 0(2), 1(6) ], [ 0(4), 1(6) ] ],[128X[104X
    [4X[28X  [ [ 0(2), 3(6) ], [ 2(4), 3(6) ] ], [ [ 0(2), 3(6) ], [ 0(4), 3(6) ] ],[128X[104X
    [4X[28X  [ [ 0(2), 5(6) ], [ 2(4), 5(6) ] ], [ [ 0(2), 5(6) ], [ 0(4), 5(6) ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
