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 Databases of Residue-Class-Wise Affine Groups and -Mappings
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<div class="ChapSects"><a href="chap6.html#X81BA344979567342">6 
 Databases of Residue-Class-Wise Affine Groups and -Mappings
</a>
<div class="ContSect"> <a href="chap6.html#X86CCBF017A746F50">6.1 The collection of examples</a>

<div class="ContSSBlock">
 <a href="chap6.html#X8714254784AFD64B">6.1-1 LoadRCWAExamples</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X85DD85DF87DE47C9">6.2 Databases of rcwa groups</a>

<div class="ContSSBlock">
 <a href="chap6.html#X793E2C5C7FC935B8">6.2-1 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</a>
 <a href="chap6.html#X7A77F7D57B08E4A5">6.2-2 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</a>
 <a href="chap6.html#X792C90B48692D0D7">6.2-3 LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X78A1A8E587C7FFD5">6.3 Databases of rcwa mappings</a>

<div class="ContSSBlock">
 <a href="chap6.html#X843E94467A1BB86C">6.3-1 LoadDatabaseOfProductsOf2ClassTranspositions</a>
 <a href="chap6.html#X85B492697ACC4A54">6.3-2 LoadDatabaseOfNonbalancedProductsOfClassTranspositions</a>
</div></div>
</div>

<h3>6 
 Databases of Residue-Class-Wise Affine Groups and -Mappings
</h3>

The RCWA package contains a number of databases of rcwa groups and rcwa mappings. They can be loaded into a GAP session by the functions described in this chapter.

<a id="X86CCBF017A746F50" name="X86CCBF017A746F50"></a>

<h4>6.1 The collection of examples</h4>

<a id="X8714254784AFD64B" name="X8714254784AFD64B"></a>

<h5>6.1-1 LoadRCWAExamples</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoadRCWAExamples</code>( )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the name of the variable to which the record containing the collection of examples of rcwa groups and -mappings loaded from the file <code class="file">pkg/rcwa/examples/examples.g</code> got bound.

The components of the examples record are records which contain the individual groups and mappings. A detailed description of some of the examples can be found in Chapter <a href="chap7.html#X7A489A5D79DA9E5C">7</a>.

<div class="example"><pre>

gap> LoadRCWAExamples();
"RCWAExamples"
gap> Set(RecNames(RCWAExamples));
[ "AbelianGroupOverPolynomialRing", "Basics", "CT3Z", "CTPZ",
 "CheckingForSolvability", "ClassSwitches",
 "ClassTranspositionProducts", "ClassTranspositionsAsCommutators",
 "CollatzFactorizationOld", "CollatzMapping", "CollatzlikePerms",
 "CoprimeMultDiv", "F2_PSL2Z", "Farkas", "FiniteQuotients",
 "FiniteVsDenseCycles", "GF2xFiniteCycles", "GrigorchukQuotients",
 "Hexagon", "HicksMullenYucasZavislak", "HigmanThompson",
 "LongCyclesOfPrimeLength", "MatthewsLeigh",
 "MaybeInfinitelyPresentedGroup", "ModuliOfPowers",
 "OddNumberOfGens_FiniteOrder", "Semilocals",
 "SlowlyContractingMappings", "Syl3_S9", "SymmetrizingCollatzTree",
 "TameGroupByCommsOfWildPerms", "Venturini", "ZxZ" ]
gap> AssignGlobals(RCWAExamples.CollatzMapping);
The following global variables have been assigned:
[ "T", "T5", "T5m", "T5p", "Tm", "Tp" ]

</pre></div>

<a id="X85DD85DF87DE47C9" name="X85DD85DF87DE47C9"></a>

<h4>6.2 Databases of rcwa groups</h4>

<a id="X793E2C5C7FC935B8" name="X793E2C5C7FC935B8"></a>

<h5>6.2-1 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</code>( )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the name of the variable to which the record containing the database of all groups generated by 3 class transpositions which interchange residue classes with moduli ≤ 6 got bound.

The database record has at least the following components (the index <code class="code">i</code> is always an integer in the range <code class="code">[1..52394]</code>, and the term <q>indices</q> always refers to list indices in that range):

<dl>
<dt><code class="code">cts</code></dt>
<dd>The list of all 69 class transpositions which interchange residue classes with moduli ≤ 6.

</dd>
<dt><code class="code">grps</code></dt>
<dd>The list of the 52394 groups -- 21948 finite and 30446 infinite ones.

</dd>
<dt><code class="code">sizes</code></dt>
<dd>The list of group orders -- it is <code class="code">Size(grps[i]) = sizes[i]</code>.

</dd>
<dt><code class="code">mods</code></dt>
<dd>The list of moduli of the groups -- it is <code class="code">Mod(grps[i]) = mods[i]</code>.

</dd>
<dt><code class="code">equalityclasses</code></dt>
<dd>A list of lists of indices <code class="code">i</code> of groups which are known to be equal, i.e. if <code class="code">i</code> and <code class="code">j</code> lie in the same list, then <code class="code">grps[i] = grps[j]</code>.

</dd>
<dt><code class="code">samegroups</code></dt>
<dd>A list of lists, where <code class="code">samegroups[i]</code> is a list of indices of groups which are known to be equal to <code class="code">grps[i]</code>.

</dd>
<dt><code class="code">conjugacyclasses</code></dt>
<dd>A list of lists of indices of groups which are known to be conjugate in RCWA(ℤ).

</dd>
<dt><code class="code">subgroups</code></dt>
<dd>A list of lists, where <code class="code">subgroups[i]</code> is a list of indices of groups which are known to be proper subgroups of <code class="code">grps[i]</code>.

</dd>
<dt><code class="code">supergroups</code></dt>
<dd>A list of lists, where <code class="code">supergroups[i]</code> is a list of indices of groups which are known to be proper supergroups of <code class="code">grps[i]</code>.

</dd>
<dt><code class="code">chains</code></dt>
<dd>A list of lists, where each list contains the indices of the groups in a descending chain of subgroups.

</dd>
<dt><code class="code">respectedpartitions</code></dt>
<dd>The list of shortest respected partitions. If <code class="code">grps[i]</code> is finite, then <code class="code">respectedpartitions[i]</code> is a list of pairs (residue, modulus) for the residue classes in the shortest respected partition <code class="code">grps[i]</code>. If <code class="code">grps[i]</code> is infinite, then <code class="code">respectedpartitions[i] = fail</code>.

</dd>
<dt><code class="code">partitionlengths</code></dt>
<dd>The list of lengths of shortest respected partitions. If the group <code class="code">grps[i]</code> is finite, then <code class="code">partitionlengths[i]</code> is the length of the shortest respected partition of <code class="code">grps[i]</code>. If <code class="code">grps[i]</code> is infinite, then <code class="code">partitionlengths[i] = 0</code>.

</dd>
<dt><code class="code">degrees</code></dt>
<dd>The list of permutation degrees, i.e. numbers of moved points, in the action of the finite groups on their shortest respected partitions. If there is no respected partition, i.e. if <code class="code">grps[i]</code> is infinite, then <code class="code">degrees[i] = 0</code>.

</dd>
<dt><code class="code">orbitlengths</code></dt>
<dd>The list of lists of orbit lengths in the action of the finite groups on their shortest respected partitions. If <code class="code">grps[i]</code> is infinite, then <code class="code">orbitlengths[i] = fail</code>.

</dd>
<dt><code class="code">permgroupgens</code></dt>
<dd>The list of lists of generators of the isomorphic permutation groups induced by the finite groups on their shortest respected partitions. If <code class="code">grps[i]</code> is infinite, then <code class="code">permgroupgens[i] = fail</code>.

</dd>
<dt><code class="code">stabilize_digitsum_base2_mod2</code></dt>
<dd>The list of indices of groups which stabilize the digit sum in base 2 modulo 2.

</dd>
<dt><code class="code">stabilize_digitsum_base2_mod3</code></dt>
<dd>The list of indices of groups which stabilize the digit sum in base 2 modulo 3.

</dd>
<dt><code class="code">stabilize_digitsum_base3_mod2</code></dt>
<dd>The list of indices of groups which stabilize the digit sum in base 3 modulo 2.

</dd>
<dt><code class="code">freeproductcandidates</code></dt>
<dd>A list of indices of groups which may be isomorphic to the free product of 3 copies of the cyclic group of order 2.

</dd>
<dt><code class="code">freeproductlikes</code></dt>
<dd>A list of indices of groups which are not isomorphic to the free product of 3 copies of the cyclic group of order 2, but where the shortest relation indicating this is relatively long.

</dd>
<dt><code class="code">abc_torsion</code></dt>
<dd>A list of pairs (index, order of product of generators) for all infinite groups for which the product of the generators has finite order.

</dd>
<dt><code class="code">cyclist</code></dt>
<dd>A list described in the comments in <code class="file">rcwa/data/3ctsgroups6/spheresizecycles.g</code>.

</dd>
<dt><code class="code">finiteorbits</code></dt>
<dd>A record described in the comments in <code class="file">rcwa/data/3ctsgroups6/finite-orbits.g</code>.

</dd>
<dt><code class="code">intransitivemodulo</code></dt>
<dd>For every modulus <code class="code">m</code> from 1 to 60, <code class="code">intransitivemodulo[m]</code> is the list of indices of groups none of whose orbits on ℤ has nontrivial intersection with all residue classes modulo <code class="code">m</code>.

</dd>
<dt><code class="code">trsstatus</code></dt>
<dd>A list of strings which describe what is known about whether the groups <code class="code">grps[i]</code> act transitively on the nonnegative integers in their support, or how the computation has failed.

</dd>
<dt><code class="code">orbitgrowthtype</code></dt>
<dd>A list of integers and lists of integers which encode what has been observed heuristically on the growth of the orbits of the groups <code class="code">grps[i]</code> on ℤ.

</dd>
</dl>
Note that the database contains an entry for every unordered 3-tuple of distinct class transpositions in <code class="code">cts</code>, which means that it contains multiple copies of equal groups -- cf. the components <code class="code">equalityclasses</code> and <code class="code">samegroups</code> described above.

To mention an example, the group <code class="code">grps[44132]</code> might be called the <q>Collatz group</q> -- its action on the set of positive integers which are not multiples of 6 is transitive if and only if the Collatz conjecture holds.

<div class="example"><pre>

gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions();
"3CTsGroups6"
gap> AssignGlobals(3CTsGroups6); # for convenience
The following global variables have been assigned:
[ "3CTsGroupsWithGivenOrbit", "Id3CTsGroup",
 "ProbablyFixesDigitSumsModulo", "ProbablyStabilizesDigitSumsModulo",
 "TriangleTypes", "abc_torsion", "chains", "conjugacyclasses", "cts",
 "cyclist", "degrees", "epifromfpgroupto_ct23z",
 "epifromfpgrouptocollatzgroup_c", "epifromfpgrouptocollatzgroup_t",
 "equalityclasses", "finiteorbits", "freeproductcandidates",
 "freeproductlikes", "groups", "grps", "intransitivemodulo",
 "minwordlengthcoprimemultdiv", "minwordlengthnonbalanced", "mods",
 "orbitgrowthtype", "orbitlengths", "partitionlengths", "permgroupgens",
 "redundant_generator", "refinementseqlngs", "respectedpartitions",
 "samegroups", "shortresidueclassorbitlengths", "sizes", "sizespos",
 "sizesset", "spheresizebound_12", "spheresizebound_24",
 "spheresizebound_4", "spheresizebound_6",
 "stabilize_digitsum_base2_mod2", "stabilize_digitsum_base2_mod3",
 "stabilize_digitsum_base3_mod2", "subgroups", "supergroups",
 "trsstatus", "trsstatuspos", "trsstatusset" ]
gap> grps[44132]; # the "3n+1 group"
<(2(3),4(6)),(1(3),2(6)),(1(2),4(6))>
gap> trsstatus[44132]; # deciding this would solve the 3n+1 problem
"exceeded memory bound"
gap> Length(Set(sizes));
1066
gap> Maximum(Filtered(sizes,IsInt)); # order of largest finite group stored
7165033589793852697531456980706732548435609645091822296777976465116824959\
2135499174617837911754921014138184155204934961004073853323458315539461543\
4480515260818409913846161473536000000000000000000000000000000000000000000\
000000
gap> PrintFactorsInt(last); 
2^200*3^103*5^48*7^28*11^16*13^13*17^8*19^6*23^6*29
gap> Positions(sizes,last); 
[ 33814, 36548 ]
gap> grps{last};
[ <(1(5),4(5)),(0(3),1(6)),(3(4),0(6))>,
 <(0(5),3(5)),(2(3),4(6)),(0(4),5(6))> ]
gap> samegroups[1]; 
[ 1, 2, 68 ]
gap> grps[1] = grps[68];
true
gap> Maximum(mods);
77760
gap> Positions(mods,last);
[ 26311, 26313, 26452, 26453, 26455, 26456, 26457, 26459, 26461, 26462,
 27781, 27784, 27785, 27786, 27788, 27789, 27790, 27791, 27829, 27832,
 30523, 30524, 30525, 30526, 30529, 30530, 30532, 30534, 32924, 32927,
 32931, 32933 ]
gap> Set(sizes{last}); 
[ 45509262704640000 ]
gap> Collected(mods);
[ [ 0, 30446 ], [ 3, 1 ], [ 4, 37 ], [ 5, 120 ], [ 6, 1450 ], [ 8, 18 ],
 [ 10, 45 ], [ 12, 3143 ], [ 15, 165 ], [ 18, 484 ], [ 20, 528 ],
 [ 24, 1339 ], [ 30, 2751 ], [ 36, 2064 ], [ 40, 26 ], [ 48, 515 ],
 [ 60, 2322 ], [ 72, 2054 ], [ 80, 44 ], [ 90, 108 ], [ 96, 108 ],
 [ 108, 114 ], [ 120, 782 ], [ 144, 310 ], [ 160, 26 ], [ 180, 206 ],
 [ 192, 6 ], [ 216, 72 ], [ 240, 304 ], [ 270, 228 ], [ 288, 14 ],
 [ 360, 84 ], [ 432, 36 ], [ 480, 218 ], [ 540, 18 ], [ 720, 120 ],
 [ 810, 112 ], [ 864, 8 ], [ 960, 94 ], [ 1080, 488 ], [ 1620, 44 ],
 [ 1920, 38 ], [ 2160, 506 ], [ 3240, 34 ], [ 3840, 12 ],
 [ 4320, 218 ], [ 4860, 16 ], [ 6480, 282 ], [ 7680, 10 ],
 [ 8640, 16 ], [ 12960, 120 ], [ 14580, 2 ], [ 25920, 34 ],
 [ 30720, 2 ], [ 38880, 12 ], [ 51840, 8 ], [ 77760, 32 ] ]
gap> Collected(trsstatus);
[ [ "> 1 orbit (mod m)", 593 ],
 [ "Mod(U DecreasingOn) exceeded <maxmod>", 23 ],
 [ "U DecreasingOn stable and exceeded memory bound", 11 ],
 [ "U DecreasingOn stable for <maxeq> steps", 5753 ],
 [ "exceeded memory bound", 497 ], [ "finite", 21948 ],
 [ "intransitive, but finitely many orbits", 8 ],
 [ "seemingly only finite orbits (long)", 1227 ],
 [ "seemingly only finite orbits (medium)", 2501 ],
 [ "seemingly only finite orbits (short)", 4816 ],
 [ "seemingly only finite orbits (very long)", 230 ],
 [ "seemingly only finite orbits (very long, very unclear)", 76 ],
 [ "seemingly only finite orbits (very short)", 208 ],
 [ "there are infinite orbits which have exponential sphere size growth"
 , 2934 ],
 [ "there are infinite orbits which have linear sphere size growth",
 10881 ],
 [ "there are infinite orbits which have unclear sphere size growth",
 86 ], [ "transitive", 562 ],
 [ "transitive up to one finite orbit", 40 ] ]

</pre></div>

<a id="X7A77F7D57B08E4A5" name="X7A77F7D57B08E4A5"></a>

<h5>6.2-2 LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions</code>( <var class="Arg">max_m</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the name of the variable to which the record containing the database of all groups generated by 3 class transpositions which interchange residue classes with moduli less than or equal to <var class="Arg">max_m</var> got bound, where <var class="Arg">max_m</var> is either 6 or 9.

If <var class="Arg">max_m</var> is 6, this is equivalent to the call of the function without argument described above. If <var class="Arg">max_m</var> is 9, the function returns a record with at least the following components (in the sequel, the indices <code class="code">i > j > k</code> are always integers in the range <code class="code">[1..264]</code>):

<dl>
<dt><code class="code">cts</code></dt>
<dd>The list of all 264 class transpositions which interchange residue classes with moduli ≤ 9.

</dd>
<dt><code class="code">mods</code></dt>
<dd>The list of moduli of the groups, i.e. <code class="code">Mod(Group(cts{[i,j,k]})) = mods[i][j][k]</code>.

</dd>
<dt><code class="code">partlengths</code></dt>
<dd>The list of lengths of shortest respected partitions of the groups in the database, i.e. <code class="code">Length(RespectedPartition(Group(cts{[i,j,k]})))</code> <code class="code">=</code> <code class="code">partlengths[i][j][k]</code>.

</dd>
<dt><code class="code">sizes</code></dt>
<dd>The list of orders of the groups, i.e. <code class="code">Size(Group(cts{[i,j,k]}))</code> <code class="code">=</code> <code class="code">sizes[i][j][k]</code>.

</dd>
<dt><code class="code">All3CTs9Indices</code></dt>
<dd>A selector function which takes as argument a function <var class="Arg">func</var> of three arguments <var class="Arg">i</var>, <var class="Arg">j</var> and <var class="Arg">k</var>. It returns a list of all triples of indices <code class="code">[<var class="Arg">i</var>,<var class="Arg">j</var>,<var class="Arg">k</var>]</code> where 264 ≥ i > j > k ≥ 1 for which <var class="Arg">func</var> returns <code class="code">true</code>.

</dd>
<dt><code class="code">All3CTs9Groups</code></dt>
<dd>A selector function which takes as argument a function <var class="Arg">func</var> of three arguments <var class="Arg">i</var>, <var class="Arg">j</var> and <var class="Arg">k</var>. It returns a list of all groups <code class="code">Group(cts{[<var class="Arg">i</var>,<var class="Arg">j</var>,<var class="Arg">k</var>]})</code> from the database for which <code class="code"><var class="Arg">func</var>(<var class="Arg">i</var>,<var class="Arg">j</var>,<var class="Arg">k</var>)</code> returns <code class="code">true</code>.

</dd>
</dl>

<div class="example"><pre>

gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions(9);
"3CTsGroups9"
gap> AssignGlobals(3CTsGroups9);
The following global variables have been assigned:
[ "All3CTs9Groups", "All3CTs9Indices", "cts", "mods", "partlengths",
 "sizes" ]
gap> PrintFactorsInt(Maximum(Filtered(Flat(sizes),n->n<>infinity)));
2^1283*3^673*5^305*7^193*11^98*13^84*17^50*19^41*23^25*29^13*31^4

</pre></div>

<a id="X792C90B48692D0D7" name="X792C90B48692D0D7"></a>

<h5>6.2-3 LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions</code>( )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the name of the variable to which the record containing the database of all groups generated by 4 class transpositions which interchange residue classes with moduli ≤ 6 for which all subgroups generated by 3 out of the 4 generators are finite got bound.

The record has at least the following components (the index <code class="code">i</code> is always an integer in the range <code class="code">[1..140947]</code>, and the term <q>indices</q> always refers to list indices in that range):

<dl>
<dt><code class="code">cts</code></dt>
<dd>The list of all 69 class transpositions which interchange residue classes with moduli ≤ 6.

</dd>
<dt><code class="code">grps4_3finite</code></dt>
<dd>The list of all 140947 groups in the database.

</dd>
<dt><code class="code">grps4_3finitepos</code></dt>
<dd>The list obtained from <code class="code">grps4_3finite</code> by replacing every group by the list of positions of its generators in the list <code class="code">cts</code>.

</dd>
<dt><code class="code">sizes4</code></dt>
<dd>The list of group orders -- it is <code class="code">Size(grps4_3finite[i]) = sizes4[i]</code>.

</dd>
<dt><code class="code">mods4</code></dt>
<dd>The list of moduli of the groups -- it is <code class="code">Mod(grps4_3finite[i]) = mods4[i]</code>.

</dd>
<dt><code class="code">conjugacyclasses4cts</code></dt>
<dd>A list of lists of indices of groups which are known to be conjugate in RCWA(ℤ).

</dd>
<dt><code class="code">grps4_3finite_reps</code></dt>
<dd>Tentative conjugacy class representatives from the list <code class="code">grps4_3finite</code> -- tentative in the sense that likely some of the groups in the list are still conjugate.

</dd>
</dl>
Note that the database contains an entry for every suitable unordered 4-tuple of distinct class transpositions in <code class="code">cts</code>, which means that it contains multiple copies of equal groups.

<div class="example"><pre>

gap> LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions(); 
"4CTsGroups6"
gap> AssignGlobals(4CTsGroups6);
The following global variables have been assigned:
[ "conjugacyclasses4cts", "cts", "grps4_3finite", "grps4_3finite_reps",
 "grps4_3finitepos", "mods4", "sizes4", "sizes4pos", "sizes4set" ]
gap> Length(grps4_3finite);
140947
gap> Length(sizes4);
140947
gap> Size(grps4_3finite[1]);
518400
gap> sizes4[1];
518400
gap> Maximum(Filtered(sizes4,IsInt));
<integer 420...000 (3852 digits)>
gap> Modulus(grps4_3finite[1]);
12
gap> mods4[1];
12
gap> Length(Set(sizes4));
7339
gap> Length(Set(mods4));
91
gap> conjugacyclasses4cts{[1..4]};
[ [ 1, 23, 563, 867 ], [ 2, 859 ], [ 3, 622 ], [ 4, 16, 868, 873 ] ]
gap> grps4_3finite[1] = grps4_3finite[23];
true
gap> grps4_3finite[4] = grps4_3finite[16];
false

</pre></div>

<a id="X78A1A8E587C7FFD5" name="X78A1A8E587C7FFD5"></a>

<h4>6.3 Databases of rcwa mappings</h4>

<a id="X843E94467A1BB86C" name="X843E94467A1BB86C"></a>

<h5>6.3-1 LoadDatabaseOfProductsOf2ClassTranspositions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoadDatabaseOfProductsOf2ClassTranspositions</code>( )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the name of the variable to which the record containing the database of products of 2 class transpositions got bound.

There are 69 class transpositions which interchange residue classes with moduli ≤ 6, thus there is a total of (69 ⋅ 68)/2 = 2346 unordered pairs of distinct such class transpositions. Looking at intersection- and subset relations between the 4 involved residue classes, we can distinguish 17 different <q>intersection types</q> (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 ≤ 6.

The components of the database record are a list <code class="code">CTPairs</code> of all 2346 pairs of distinct class transpositions which interchange residue classes with moduli ≤ 6, functions <code class="code">CTPairsIntersectionTypes</code>, <code class="code">CTPairIntersectionType</code> and <code class="code">CTPairProductType</code>, as well as data lists <code class="code">OrdersMatrix</code>, <code class="code">CTPairsProductClassification</code>, <code class="code">CTPairsProductType</code>, <code class="code">CTProds12</code> and <code class="code">CTProds32</code>. -- For the description of these components, see the file <code class="file">pkg/rcwa/data/ctproducts/ctprodclass.g</code>.

<div class="example"><pre>

gap> LoadDatabaseOfProductsOf2ClassTranspositions();
"CTProducts"
gap> Set(RecNames(CTProducts));
[ "CTPairIntersectionType", "CTPairProductType", "CTPairs",
 "CTPairsIntersectionTypes", "CTPairsProductClassification",
 "CTPairsProductType", "CTProds12", "CTProds32", "OrdersMatrix" ]
gap> Length(CTProducts.CTPairs);
2346
gap> Collected(List(CTProducts.CTPairsProductType,l->l[2])); # order stats
[ [ 2, 165 ], [ 3, 255 ], [ 4, 173 ], [ 6, 693 ], [ 10, 2 ],
 [ 12, 345 ], [ 15, 4 ], [ 20, 10 ], [ 30, 120 ], [ 60, 44 ],
 [ infinity, 535 ] ]

</pre></div>

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<h5>6.3-2 LoadDatabaseOfNonbalancedProductsOfClassTranspositions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoadDatabaseOfNonbalancedProductsOfClassTranspositions</code>( )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the name of the variable to which the record containing the database of non-balanced products of class transpositions got bound.

This database contains a list of the 24 pairs of class transpositions which interchange residue classes with moduli ≤ 6 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.

<div class="example"><pre>

gap> LoadDatabaseOfNonbalancedProductsOfClassTranspositions();
"CTProductsNB"
gap> Set(RecNames(CTProductsNB));
[ "PairsOfCTsWhoseProductIsNotBalanced",
 "TriplesOfCTsWhoseProductHasCoprimeMultiplierAndDivisor" ]
gap> CTProductsNB.PairsOfCTsWhoseProductIsNotBalanced;
[ [ ( 1(2), 2(4) ), ( 2(4), 3(6) ) ], [ ( 1(2), 2(4) ), ( 2(4), 5(6) ) ],
 [ ( 1(2), 2(4) ), ( 2(4), 1(6) ) ], [ ( 1(2), 0(4) ), ( 0(4), 1(6) ) ],
 [ ( 1(2), 0(4) ), ( 0(4), 3(6) ) ], [ ( 1(2), 0(4) ), ( 0(4), 5(6) ) ],
 [ ( 0(2), 1(4) ), ( 1(4), 2(6) ) ], [ ( 0(2), 1(4) ), ( 1(4), 4(6) ) ],
 [ ( 0(2), 1(4) ), ( 1(4), 0(6) ) ], [ ( 0(2), 3(4) ), ( 3(4), 4(6) ) ],
 [ ( 0(2), 3(4) ), ( 3(4), 2(6) ) ], [ ( 0(2), 3(4) ), ( 3(4), 0(6) ) ],
 [ ( 1(2), 2(6) ), ( 3(4), 2(6) ) ], [ ( 1(2), 2(6) ), ( 1(4), 2(6) ) ],
 [ ( 1(2), 4(6) ), ( 3(4), 4(6) ) ], [ ( 1(2), 4(6) ), ( 1(4), 4(6) ) ],
 [ ( 1(2), 0(6) ), ( 1(4), 0(6) ) ], [ ( 1(2), 0(6) ), ( 3(4), 0(6) ) ],
 [ ( 0(2), 1(6) ), ( 2(4), 1(6) ) ], [ ( 0(2), 1(6) ), ( 0(4), 1(6) ) ],
 [ ( 0(2), 3(6) ), ( 2(4), 3(6) ) ], [ ( 0(2), 3(6) ), ( 0(4), 3(6) ) ],
 [ ( 0(2), 5(6) ), ( 2(4), 5(6) ) ], [ ( 0(2), 5(6) ), ( 0(4), 5(6) ) ]
]

</pre></div>

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