  
  [1X9 [33X[0;0YOrbit enumeration by suborbits[133X[101X
  
  [33X[0;0YThe  code  described  in  this  chapter  is quite complicated and one has to
  understand  quite  a  lot of theory to use it. The reason for this is that a
  lot  of  preparatory  data has to be found and supplied by the user in order
  for this code to run at all. Also the situations in which it can be used are
  quite  special.  However,  in  such  a  situation, the user is rewarded with
  impressive performance.[133X
  
  [33X[0;0YThe  main  reference  for the theory is [MNW07]. We briefly recall the basic
  setup:  Let  [22XG[122X  be  a  group acting from the right on some set [22XX[122X. Let [22Xk[122X be a
  natural number, set [22XX_{k+1} := X[122X, and let[133X
  
  
  [24X[33X[0;6YU_1 < U_2 < \ldots < U_k < U_{{k+1}} = G[133X
  
  [124X
  
  [33X[0;0Ybe  a chain of [21Xhelper[121X subgroups. Further, for [22X1 ≤ i ≤ k[122X let [22XX_i[122X be a [22XU_i[122X set
  and let [22Xπ_i : X_{i+1} -> X_i[122X be a homomorphism of [22XU_i[122X-sets.[133X
  
  [33X[0;0YThis chapter starts with a section about the main orbit enumeration function
  and the corresponding preparation functions. It then proceeds with a section
  on  the  used  data  structures, which will necessarily be rather technical.
  Finally,  the  chapter  concludes  with  a  section  on  higher  level  data
  structures like lists of orbit-by-suborbit objects and their administration.
  Note that there are quite a few examples in Chapter [14X11[114X.[133X
  
  
  [1X9.1 [33X[0;0Y[10XOrbitBySuborbits[110X[101X[1X and its resulting objects[133X[101X
  
  [1X9.1-1 OrbitBySuborbit[101X
  
  [33X[1;0Y[29X[2XOrbitBySuborbit[102X( [3Xsetup[103X, [3Xp[103X, [3Xj[103X, [3Xl[103X, [3Xi[103X, [3Xpercentage[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Yan orbit-by-suborbit object[133X
  
  [33X[0;0YThis  is  the  main  function  in the whole business. All notations from the
  beginning  of  this  Chapter [14X9[114X remain in place. The argument [3Xsetup[103X must be a
  setup record lying in the filter [2XIsOrbitBySuborbitSetup[102X ([14X9.3-1[114X) described in
  detail     in     Section     [14X9.3[114X    and    produced    for    example    by
  [2XOrbitBySuborbitBootstrapForVectors[102X                 ([14X9.2-1[114X)                or
  [2XOrbitBySuborbitBootstrapForLines[102X  ([14X9.2-2[114X) described below. In particular, it
  contains  all  the  generators  for [22XG[122X and the helper subgroups acting on the
  various  sets.  The argument [3Xp[103X must be the starting point of the orbit. Note
  that  the  function  possibly  does  not take [3Xp[103X itself as starting point but
  rather  its [22XU_k[122X-minimalisation, which is a point in the same [22XU_k[122X-orbit as [3Xp[103X.
  This  information  is  important  for  the  resulting  stabiliser  and words
  representing the [22XU_k[122X-suborbits.[133X
  
  [33X[0;0YThe integers [3Xj[103X, [3Xl[103X, and [3Xi[103X, for which [22Xk+1 ≥ [3Xj[103X ≥ [3Xl[103X > [3Xi[103X ≥ 1[122X must hold, determine
  the  running mode. [3Xj[103X indicates in which set [22XX_j[122X the point [3Xp[103X lies and thus in
  which  set  the  orbit  enumeration  takes  place, with [22Xj=k+1[122X indicating the
  original  set  [22XX[122X.  The  value  [3Xl[103X  indicates  which  group  to  use for orbit
  enumeration.  So  the  result  will  be a [22XU_l[122X orbit, with [22X[3Xl[103X=[3Xk[103X+1[122X indicating a
  [22XG[122X-orbit.  Finally,  the  value  [3Xi[103X  indicates  which  group to use for the [21Xby
  suborbit[121X  part,  that  is,  the orbit will be enumerated [21Xby [22XU_[3Xi[103X[122X-orbits[121X. Note
  that  nearly  all  possible combinations of these parameters actually occur,
  because  this  function  is  also  used  in  the  [21Xon-the-fly[121X  precomputation
  happening  behind the scenes. The most common usage of this function for the
  user is [22X[3Xj[103X=[3Xl[103X=[3Xk[103X+1[122X and [22X[3Xi[103X=k[122X.[133X
  
  [33X[0;0YFinally,  the  integer [3Xpercentage[103X says, how much of the full orbit should be
  enumerated, the value is in percent, thus [22X100[122X means the full orbit. Usually,
  only  values  greater  than  [22X50[122X are sensible, because one can only prove the
  size of the orbit after enumerating at least half of it.[133X
  
  [33X[0;0YThe  result is an [21Xorbit-by-suborbit[121X object. For such an object in particular
  the   operations   [2XSize[102X   ([14X9.1-3[114X),   [2XSeed[102X   ([14X9.1-4[114X),   [2XSuborbitsDb[102X  ([14X9.1-5[114X),
  [2XWordsToSuborbits[102X  ([14X9.1-6[114X),  [2XMemory[102X  ([14X9.1-7[114X),  [2XStabilizer[102X  ([14X9.1-8[114X),  and [2XSeed[102X
  ([14X9.1-4[114X) are defined, see below.[133X
  
  [1X9.1-2 OrbitBySuborbitKnownSize[101X
  
  [33X[1;0Y[29X[2XOrbitBySuborbitKnownSize[102X( [3Xsetup[103X, [3Xp[103X, [3Xj[103X, [3Xl[103X, [3Xi[103X, [3Xpercentage[103X, [3Xknownsize[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Yan orbit-by-suborbit object[133X
  
  [33X[0;0YBasically  does the same as [2XOrbitBySuborbit[102X ([14X9.1-1[114X) but does not compute the
  stabiliser  by  evaluating Schreier words. Instead, the size of the orbit to
  enumerate  must already be known and be given in the argument [3Xknownsize[103X. The
  other arguments are as for the function [2XOrbitBySuborbit[102X ([14X9.1-1[114X).[133X
  
  [1X9.1-3 Size[101X
  
  [33X[1;0Y[29X[2XSize[102X( [3Xorb[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the number of points in the orbit-by-suborbit [3Xorb[103X.[133X
  
  [1X9.1-4 Seed[101X
  
  [33X[1;0Y[29X[2XSeed[102X( [3Xorb[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya point in the orbit[133X
  
  [33X[0;0YReturns  the  starting  point  of  the  orbit-by-suborbit  [3Xorb[103X.  It  is  the
  [22XU_i[122X-minimalisation of the starting point given to [2XOrbitBySuborbit[102X ([14X9.1-1[114X).[133X
  
  [1X9.1-5 SuborbitsDb[101X
  
  [33X[1;0Y[29X[2XSuborbitsDb[102X( [3Xorb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya database of suborbits[133X
  
  [33X[0;0YReturns  the  data base of suborbits of the orbit-by-suborbit object [3Xorb[103X. In
  particular,  such  a  database  object has methods for the operations [2XMemory[102X
  ([14X9.1-7[114X),   [2XTotalLength[102X   ([14X9.1-11[114X),   and   [2XRepresentatives[102X   ([14X9.1-12[114X).   For
  descriptions see below.[133X
  
  [1X9.1-6 WordsToSuborbits[101X
  
  [33X[1;0Y[29X[2XWordsToSuborbits[102X( [3Xorb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of words[133X
  
  [33X[0;0YReturns  a list of words in the groups [22XU_*[122X reaching each of the suborbits in
  the  orbit-by-suborbit  [3Xorb[103X.  Here  a  word  is a list of integers. Positive
  numbers  index  generators in following numbering: The first few numbers are
  numbers  of  generators  of  [22XU_1[122X  the  next  few  adjacent numbers index the
  generators  of  [22XU_2[122X and so on until the generators of [22XG[122X in the end. Negative
  numbers indicate the corresponding inverses of these generators.[133X
  
  [33X[0;0YNote  that  [2XOrbitBySuborbit[102X  ([14X9.1-1[114X)  takes  the  [22XU_i[122X-minimalisation  of the
  starting  point as its starting point and the words here are all relative to
  this new starting point.[133X
  
  [1X9.1-7 Memory[101X
  
  [33X[1;0Y[29X[2XMemory[102X( [3Xob[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the amount of memory needed by the object [3Xob[103X, which can be either an
  orbit-by-suborbit  object,  a  suborbit database object, or an object in the
  filter [2XIsOrbitBySuborbitSetup[102X ([14X9.3-1[114X). The amount of memory used is given in
  bytes.  Note  that this includes all hashes, databases, and preparatory data
  of  substantial  size.  For  orbit-by-suborbits  the  memory  needed for the
  precomputation is not included, ask the setup object for that.[133X
  
  [1X9.1-8 Stabilizer[101X
  
  [33X[1;0Y[29X[2XStabilizer[102X( [3Xorb[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya permutation group[133X
  
  [33X[0;0YReturns the stabiliser of the starting point of the orbit-by-suborbit in [3Xorb[103X
  in  form  of  a  permutation  group,  using  the  given faithful permutation
  representation in the setup record.[133X
  
  [33X[0;0YNote  that  [2XOrbitBySuborbit[102X  ([14X9.1-1[114X)  takes  the  [22XU_i[122X-minimalisation  of the
  starting point as its starting point and the stabiliser returned here is the
  one of this new starting point.[133X
  
  [1X9.1-9 StabWords[101X
  
  [33X[1;0Y[29X[2XStabWords[102X( [3Xorb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of words[133X
  
  [33X[0;0YReturns  generators  for  the  stabiliser  of  the  starting  point  of  the
  orbit-by-suborbit  in  [3Xorb[103X  in form of words as described with the operation
  [2XWordsToSuborbits[102X  ([14X9.1-6[114X). Note again that [2XOrbitBySuborbit[102X ([14X9.1-1[114X) takes the
  [22XU_i[122X-minimalisation  of  the  starting  point  as  its starting point and the
  stabiliser returned here is the one of this new starting point.[133X
  
  [1X9.1-10 SavingFactor[101X
  
  [33X[1;0Y[29X[2XSavingFactor[102X( [3Xorb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns   the  quotient  of  the  total  number  of  points  stored  in  the
  orbit-by-suborbit  [3Xorb[103X and the total number of [22XU[122X-minimal points stored. Note
  that the memory for the precomputations is not considered here![133X
  
  [33X[0;0YThe following operations apply to orbit-by-suborbit database objects:[133X
  
  [1X9.1-11 TotalLength[101X
  
  [33X[1;0Y[29X[2XTotalLength[102X( [3Xdb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns  the  total  number  of  points  stored  in  all  suborbits  in  the
  orbit-by-suborbit database [3Xdb[103X.[133X
  
  [1X9.1-12 Representatives[101X
  
  [33X[1;0Y[29X[2XRepresentatives[102X( [3Xdb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of points[133X
  
  [33X[0;0YReturns   a   list  of  representatives  of  the  suborbits  stored  in  the
  orbit-by-suborbit database [3Xdb[103X.[133X
  
  [1X9.1-13 SavingFactor[101X
  
  [33X[1;0Y[29X[2XSavingFactor[102X( [3Xdb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns  the  quotient  of the total number of points stored in the suborbit
  database  [3Xdb[103X  and the total number of [22XU[122X-minimal points stored. Note that the
  memory for the precomputations is not considered here![133X
  
  [1X9.1-14 OrigSeed[101X
  
  [33X[1;0Y[29X[2XOrigSeed[102X( [3Xorb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya point[133X
  
  [33X[0;0YReturns the original starting point for the orbit, not yet minimalised.[133X
  
  
  [1X9.2 [33X[0;0YPreparation functions for [2XOrbitBySuborbit[102X[101X[1X ([14X9.1-1[114X)[133X[101X
  
  [1X9.2-1 OrbitBySuborbitBootstrapForVectors[101X
  
  [33X[1;0Y[29X[2XOrbitBySuborbitBootstrapForVectors[102X( [3Xgens[103X, [3Xpermgens[103X, [3Xsizes[103X, [3Xcodims[103X, [3Xopt[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya setup record in the filter [2XIsOrbitBySuborbitSetup[102X ([14X9.3-1[114X)[133X
  
  [33X[0;0YAll  notations  from  the  beginning of this Chapter [14X9[114X remain in place. This
  function  is  for  the  action of matrices on row vectors, so all generators
  must  be matrices. The set [22XX[122X thus is a row space usually over a finite field
  and  the sets [22XX_i[122X are quotient spaces. The matrix generators for the various
  groups  have  to  be  adjusted  with  a base change, such that the canonical
  projection onto [22XX_i[122X is just to take the first few entries in a vector, which
  means,  that  the  submodules divided out are generated by the last standard
  basis vectors.[133X
  
  [33X[0;0YThe  first  argument  [3Xgens[103X  must be a list of lists of generators. The outer
  list  must  have length [22Xk+1[122X with entry [22Xi[122X being a list of matrices generating
  [22XU_i[122X,  given in the action on [22XX=X_{k+1}[122X. The above mentioned base change must
  have  been done. The second argument [3Xpermgens[103X must be an analogous list with
  generator  lists  for  the  [22XU_i[122X.  These  representations are used to compute
  membership  and  group orders of stabilisers. In its simplest form, [3Xpermgens[103X
  is a list of permutation representations of the same degree, giving a set of
  generators  for each individual group [22XU_i[122X. Alternatively, if for some [22XU_i[122X, [22Xi
  > 1[122X, it is required that the stabilizer of its action is to be calculated as
  a  matrix  group,  generators  of  [22XU_i[122X  in some matrix representation may be
  supplied.  However,  it  is  then  mandatory  that  for  all [22X1 < i ≤ k+1[122X the
  generator  lists  have the following format: The [22Xi[122X-th entry of [3Xpermgens[103X is a
  list  concatenating the generator lists of [22XU_1[122X up to [22XU_i[122X (in this order) all
  of   whose   elements   are  in  either  some  permutation  or  some  matrix
  representation. Note that currently, the generators of [22XU_1[122X need to be always
  given  in a permutation representation. The argument [3Xsizes[103X must be a list of
  length  [22Xk+1[122X  and  entry [22Xi[122X must be the group order of [22XU_i[122X (again with [22XU_{k+1}[122X
  being [22XG[122X). Finally, the argument [3Xcodims[103X must be a list of length [22Xk[122X containing
  integers  with  the  [22Xi[122Xth  entry  being  the codimension of the [22XU_i[122X-invariant
  subspace [22XY_i[122X of [22XX[122X with [22XX_i = X/Y_i[122X. These codimensions must not decrease for
  obvious  reasons, but some of them may be equal. The last argument [3Xopt[103X is an
  options record. See below for possible entries.[133X
  
  [33X[0;0YThe function does all necessary steps to fill a setup record (see [14X9.3[114X) to be
  used with [2XOrbitBySuborbit[102X ([14X9.1-1[114X). For details see the code.[133X
  
  [33X[0;0YCurrently,  the  following  components  in  the  options  record  [3Xopt[103X have a
  meaning:[133X
  
  [8X[10Xregvecfachints[110X[8X[108X
        [33X[0;6YIf  bound it must be a list. In position [22Xi[122X for [22Xi>1[122X there may be a list
        of  vectors  in  the  [22Xi[122X-th  quotient  space  [22XX_i[122X  that  can be used to
        distinguish  the  left [22XU_{i-1}[122X cosets in [22XU_i[122X. All vectors in this list
        are tried and the first one that actually works is used.[133X
  
  [8X[10Xregvecfullhints[110X[8X[108X
        [33X[0;6YIf  bound it must be a list. In position [22Xi[122X for [22Xi>1[122X there may be a list
        of  vectors  in  the  full space [22XX[122X that can be used to distinguish the
        left [22XU_{i-1}[122X cosets in [22XU_i[122X. All vectors in this list are tried and the
        first one that actually works is used.[133X
  
  [8X[10Xstabchainrandom[110X[8X[108X
        [33X[0;6YIf bound the value is copied into the [10Xstabchainrandom[110X component of the
        setup record.[133X
  
  [8X[10Xnostabchainfullgroup[110X[8X[108X
        [33X[0;6YIf bound it must be [9Xtrue[109X or [9Xfalse[109X. If it is unbound or set to [9Xtrue[109X, no
        stabilizer chain is computed for the group [22XU_k+1[122X. Its default value is
        [9Xfalse[109X.[133X
  
  [1X9.2-2 OrbitBySuborbitBootstrapForLines[101X
  
  [33X[1;0Y[29X[2XOrbitBySuborbitBootstrapForLines[102X( [3Xgens[103X, [3Xpermgens[103X, [3Xsizes[103X, [3Xcodims[103X, [3Xopt[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya setup record in the filter [2XIsOrbitBySuborbitSetup[102X ([14X9.3-1[114X)[133X
  
  [33X[0;0YAll  notations  from  the  beginning of this Chapter [14X9[114X remain in place. This
  does  exactly  the same as [2XOrbitBySuborbitBootstrapForVectors[102X ([14X9.2-1[114X) except
  that  it  handles  the case of matrices acting on one-dimensional subspaces.
  Those one-dimensional subspaces are represented by normalised vectors, where
  a vector is normalised if its first non-vanishing entry is equal to [22X1[122X.[133X
  
  [1X9.2-3 OrbitBySuborbitBootstrapForSpaces[101X
  
  [33X[1;0Y[29X[2XOrbitBySuborbitBootstrapForSpaces[102X( [3Xgens[103X, [3Xpermgens[103X, [3Xsizes[103X, [3Xcodims[103X, [3Xspcdim[103X, [3Xopt[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya setup record in the filter [2XIsOrbitBySuborbitSetup[102X ([14X9.3-1[114X)[133X
  
  [33X[0;0YAll  notations  from  the  beginning of this Chapter [14X9[114X remain in place. This
  does  exactly  the same as [2XOrbitBySuborbitBootstrapForVectors[102X ([14X9.2-1[114X) except
  that it handles the case of matrices acting on [3Xspcdim[103X-dimensional subspaces.
  Those subspaces are represented by fully echelonised bases.[133X
  
  
  [1X9.3 [33X[0;0YData structures for orbit-by-suborbits[133X[101X
  
  [33X[0;0YThe  description  in this section is necessarily technical. It is meant more
  as  extended  annotations  to  the  source  code than as user documentation.
  Usually  it  should  not  be  necessary  for  the  user  to know the details
  presented  here.  The  function [2XOrbitBySuborbit[102X ([14X9.1-1[114X) needs an information
  record of the following form:[133X
  
  [1X9.3-1 IsOrbitBySuborbitSetup[101X
  
  [33X[1;0Y[29X[2XIsOrbitBySuborbitSetup[102X( [3Xob[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YObjects  in  this  category  are  also in [10XIsComponentObjRep[110X. We describe the
  components, refering to the setup at the beginning of this Chapter [14X9[114X.[133X
  
  [8X[10Xk[110X[8X[108X
        [33X[0;6YThe number of helper subgroups.[133X
  
  [8X[10Xsize[110X[8X[108X
        [33X[0;6YA  list  of  length  [22Xk+1[122X  containing  the  orders  of  the groups [22XU_i[122X,
        including [22XU_{k+1} = G[122X.[133X
  
  [8X[10Xindex[110X[8X[108X
        [33X[0;6YA  list  of length [22Xk[122X with the index [22X[U_i:U_{i-1}][122X in position [22Xi[122X ([22XU_0 =
        {1}[122X).[133X
  
  [8X[10Xels[110X[8X[108X
        [33X[0;6YA  list  of  length  [22Xk+1[122X  containing generators of the groups in their
        action  on various sets. In position [22Xi[122X we store all the generators for
        all  groups acting on [22XX_i[122X, that is for the groups [22XU_1, ..., U_i[122X (where
        position  [22Xk+1[122X  includes  the  generators  for  [22XG[122X. In each position the
        generators of all those groups are concatentated starting with [22XU_1[122X and
        ending with [22XU_i[122X.[133X
  
  [8X[10Xelsinv[110X[8X[108X
        [33X[0;6YThe  inverses  of  all  the  elements in the [10Xels[110X component in the same
        arrangement.[133X
  
  [8X[10Xtrans[110X[8X[108X
        [33X[0;6YA  list  of  length  [22Xk[122X  in which position [22Xi[122X for [22Xi>1[122X contains a list of
        words  in the generators for a transversal of [22XU_{i-1}[122X in [22XU_i[122X (with [22XU_0
        = 1[122X).[133X
  
  [8X[10Xpifunc[110X[8X[108X
        [33X[0;6YProjection  functions.  This  is  a  list  of length [22Xk+1[122X containing in
        position  [22Xj[122X  a  list  of  length  [22Xj-1[122X  containing  in position [22Xi[122X a [5XGAP[105X
        function doing the projection [22XX_j -> X_i[122X. These [5XGAP[105X functions take two
        arguments,  namely  the  point to map and secondly the value of the [10Xpi[110X
        component  at  positions  [10X[j][i][110X.  Usually  [10Xpifunc[110X is just the slicing
        operator  in  [5XGAP[105X  and [10Xpi[110X contains the components to project onto as a
        range object.[133X
  
  [8X[10Xpi[110X[8X[108X
        [33X[0;6YSee the description of the [10Xpifunc[110X component.[133X
  
  [8X[10Xop[110X[8X[108X
        [33X[0;6YA  list  of  [22Xk+1[122X  [5XGAP[105X operation functions, each taking a point [22Xp[122X and a
        generator [22Xg[122X in the action given by the index and returning [22Xpg[122X.[133X
  
  [8X[10Xinfo[110X[8X[108X
        [33X[0;6YA  list  of  length  [22Xk[122X containing a hash table with the minimalisation
        lookup  data.  These  hash  tables  grow  during orbit enumerations as
        precomputations are done behind the scenes.[133X
  
        [33X[0;6Y[10Xinfo[1][110X  contains  precomputation  data  for [22XX_1[122X. Assume [22Xx ∈ X_1[122X to be
        [22XU_1[122X-minimal.  For  all  [22Xz  ∈ xU_1[122X with [22Xz ≠ x[122X we store the number of an
        element  in  the  [10Xwordcache[110X  mapping [22Xz[122X to [22Xx[122X. For [22Xz=x[122X we store a record
        with  two  components  [10Xgens[110X and [10Xsize[110X, where [10Xgens[110X stores generators for
        the stabiliser Stab[22X_{U_1}(x)[122X as words in the group generators and [10Xsize[110X
        stores the size of that stabiliser.[133X
  
        [33X[0;6Y[10Xinfo[i][110X  for  [22Xi>1[122X contains precomputation data for [22XX_i[122X. Assume [22Xx ∈ X_i[122X
        to  be  [22XU_i[122X-minimal.  For  all  [22XU_{i-1}[122X-minimal [22Xz ∈ xU_i ∖ xU_{i-1}[122X we
        store  the  number  of an element in [10Xtrans[i][110X mapping [22Xz[122X into [22XxU_{i-1}[122X.
        For  all [22XU_{i-1}[122X-minimal [22Xz ∈ xU_{i-1}[122X with [22Xz ≠ x[122X we store the negative
        of  the  number  of  a  word in [10Xwordcache[110X that is in the generators of
        [22XU_{i-1}[122X  and  maps [22Xz[122X to [22Xx[122X. For [22Xz=x[122X we store the stabiliser information
        as in the case [22Xi=1[122X.[133X
  
        [33X[0;6YThis  information  together  with  the  information  in  the following
        componente allows the minimalisation function to do its job.[133X
  
  [8X[10Xcosetrecog[110X[8X[108X
        [33X[0;6YA list of length [22Xk[122X beginning with the index [22X1[122X. The entry at position [22Xi[122X
        is  bound to a function taking [22X3[122X arguments, namely [22Xi[122X itself, a word in
        the group generators of [22XU_1, ..., U_k[122X which lies in [22XU_i[122X, and the setup
        record.  The function computes the number [10Xj[110X of an element in [10Xtrans[i][110X,
        such that the element of [22XU_i[122X described by the word lies in [10Xtrans[i][j]
        U_{{i-1}}[110X.[133X
  
  [8X[10Xcosetinfo[110X[8X[108X
        [33X[0;6YA list of things that can be used by the functions in [10Xcosetrecog[110X.[133X
  
  [8X[10Xsuborbnr[110X[8X[108X
        [33X[0;6YA  list  of  length  [22Xk[122X  that  contains  in  position  [22Xi[122X  the number of
        [22XU_i[122X-orbits in [22XX_i[122X archived in [10Xinfo[i][110X during precomputation.[133X
  
  [8X[10Xsumstabl[110X[8X[108X
        [33X[0;6YA  list  of  length [22Xk[122X that contains in position [22Xi[122X the sum of the point
        stabiliser  sizes  of  all  [22XU_i[122X-orbits  [22XX_i[122X archived in [10Xinfo[i][110X during
        precomputation.[133X
  
  [8X[10Xpermgens[110X[8X[108X
        [33X[0;6YA list of length [22Xk+1[122X containing in position [22Xi[122X generators for [22XU_1, ...,
        U_i[122X in a faithful permutation representation of [22XU_i[122X. Generators fit to
        the  generators  in  [10Xels[110X.  For  the  variant  [2XOrbitBySuborbitKnownSize[102X
        ([14X9.1-2[114X) the [22Xk+1[122X entry can be unbound.[133X
  
  [8X[10Xpermgensinv[110X[8X[108X
        [33X[0;6YThe inverses of the generators in [10Xpermgens[110X in the same arrangement.[133X
  
  [8X[10Xsample[110X[8X[108X
        [33X[0;6YA list of length [22Xk+1[122X containing sample points in the sets [22XX_i[122X.[133X
  
  [8X[10Xstabchainrandom[110X[8X[108X
        [33X[0;6YThe  value  is  used  as the value for the [10Xrandom[110X option for [10XStabChain[110X
        calculations  to  determine stabiliser sizes. Note that the algorithms
        are randomized if you use this feature with a value smaller than [22X1000[122X.[133X
  
  [8X[10Xwordhash[110X[8X[108X
        [33X[0;6YA  hash to quickly recognise already used words. For every word in the
        hash  the  position  of  that  word in the [10Xwordcache[110X list is stored as
        value in the hash.[133X
  
  [8X[10Xwordcache[110X[8X[108X
        [33X[0;6YA list of words in the wordcache for indexing purposes.[133X
  
  [8X[10Xhashlen[110X[8X[108X
        [33X[0;6YInitial  length  of  hash  tables used for the enumeration of lists of
        [22XU_i[122X-minimal points.[133X
  
  [8X[10Xstaborblenlimit[110X[8X[108X
        [33X[0;6YThis  contains  the  limit,  up  to  which  orbits  of stabilisers are
        computed  using word action. After this limit, the stabiliser elements
        are actually evaluated in the group.[133X
  
  [8X[10Xstabsizelimitnostore[110X[8X[108X
        [33X[0;6YIf  the  stabiliser  in  the  quotient  is larger than this limit, the
        suborbit is not stored.[133X
  
  [8X[10Xcache[110X[8X[108X
        [33X[0;6YA  linked  list  cache  object  (see [2XLinkedListCache[102X ([14X5.2-1[114X)) to store
        already  computed transversal elements. The cache nodes are referenced
        in the [10Xtranscache[110X component and are stored in the cache [10Xcache[110X.[133X
  
  [8X[10Xtranscache[110X[8X[108X
        [33X[0;6YThis  is  a  list  of  lists of weak pointer objects. The weak pointer
        object   at  position  [10X[i][j][110X  holds  references  to  cache  nodes  of
        transversal elements of [22XU_i-1[122X in [22XU_i[122X in representation [22Xj[122X.[133X
  
  
  [1X9.3-2 [33X[0;0YThe global record [10XORB[110X[101X[1X[133X[101X
  
  [33X[0;0YIn  this  section  we  describe  the  global record [10XORB[110X, which contains some
  entries  that can tune the behaviour of the orbit-by-suborbit functions. The
  record has the following components:[133X
  
  [8X[10XMINSHASHLEN[110X[8X[108X
        [33X[0;6YThis  positive  integer  is  the  initial  value of the hash size when
        enumerating orbits of stored stabilisers to find all or search through
        [22XU_{i-1}[122X-minimal vectors in an [22XU_i[122X-orbit. The default value is [22X1000[122X.[133X
  
  [8X[10XORBITBYSUBORBITDEPTH[110X[8X[108X
        [33X[0;6YThis    integer    indicates    how    many    recursive    calls   to
        [10XOrbitBySubOrbitInner[110X  have  been  done.  The  initial  value  is  [22X0[122X to
        indicate  that  no  such call has happened. This variable is necessary
        since  the  minimalisation routine sometimes uses [10XOrbitBySubOrbitInner[110X
        recursively  to  complete  some  precomputation [21Xon the fly[121X during some
        other orbit-by-suborbit enumeration. This component is always set to [22X0[122X
        automatically     when     calling    [2XOrbitBySuborbit[102X    ([14X9.1-1[114X)    or
        [2XOrbitBySuborbitKnownSize[102X  ([14X9.1-2[114X)  so the user should usually not have
        to worry about it at all.[133X
  
  [8X[10XPATIENCEFORSTAB[110X[8X[108X
        [33X[0;6YThis integer indicates how many Schreier generators for the stabiliser
        are  tried before assuming that the stabiliser is complete. Whenever a
        new  generator  for the stabiliser is found that increases the size of
        the  currently known stabiliser, the count is reset to [22X0[122X that is, only
        when  [10XORB.PATIENCEFORSTAB[110X  unsuccessful  Schreier generators have been
        tried  no  more Schreier generators are created. The default value for
        this  component  is  [22X1000[122X.  This  feature  is  purely  heuristical and
        therefore this value has to be adjusted for some orbit enumerations.[133X
  
  [8X[10XPLEASEEXITNOW[110X[8X[108X
        [33X[0;6YThis value is usually set to [9Xfalse[109X. Setting it to [9Xtrue[109X in a break loop
        tells  the  orbit-by-suborbit  routines to exit gracefully at the next
        possible  time.  Simply  leaving  such  a break loop with [9Xquit;[109X is not
        safe,  since  the  routines  might  be  in  the  process  of  updating
        precomputation  data  and  the  data structures might be left corrupt.
        Always use this component to leave an orbit enumeration prematurely.[133X
  
  [8X[10XREPORTSUBORBITS[110X[8X[108X
        [33X[0;6YThis  positive  integer  governs  how often information messages about
        newly  found  suborbits  are printed. The default value is [22X1000[122X saying
        that  after  every  [22X1000[122X  suborbits  a message is printed, if the info
        level  is at its default value [22X1[122X. If the info level is increased, then
        this  component  does  no  longer  affect  the  printing and all found
        suborbits are reported.[133X
  
  [8X[10XTRIESINQUOTIENT[110X[8X and [10XTRIESINWHOLESPACE[110X[8X[108X
        [33X[0;6YThe  bootstrap  routines  [2XOrbitBySuborbitBootstrapForVectors[102X  ([14X9.2-1[114X),
        [2XOrbitBySuborbitBootstrapForLines[102X              ([14X9.2-2[114X)              and
        [2XOrbitBySuborbitBootstrapForSpaces[102X   ([14X9.2-3[114X)   all   need   to  compute
        transversals  of  one  helper subgroup in the next one. They use orbit
        enumerations   in  various  spaces  to  achieve  this.  The  component
        [10XTRIESINQUOTIENT[110X must be a non-negative integer and indicates how often
        a  random  vector in the corresponding quotient space is tried to find
        an  orbit  that  can  distinguish  between cosets. The other component
        [10XTRIESINWHOLESPACE[110X  also  must  be a non-negative integer and indicates
        how  often  a  random  vector in the whole space is tried. The default
        values are [22X3[122X and [22X20[122X resepectively.[133X
  
  
  [1X9.4 [33X[0;0YLists of orbit-by-suborbit objects[133X[101X
  
  [33X[0;0YThere   are   a   few   functions   that   help  to  administrate  lists  of
  orbit-by-suborbits.[133X
  
  [1X9.4-1 InitOrbitBySuborbitList[101X
  
  [33X[1;0Y[29X[2XInitOrbitBySuborbitList[102X( [3Xsetup[103X, [3Xnrrandels[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list of orbit-by-suborbits object[133X
  
  [33X[0;0YCreates  an  object  that  stores a list of orbit-by-suborbits. The argument
  [3Xsetup[103X  must  be  an  orbit-by-suborbit setup record and [3Xnrrandels[103X must be an
  integer.  It  indicates how many random elements in [22XG[122X should be used to do a
  probabilistic  check  for  membership  in  case an orbit-by-suborbit is only
  partially known.[133X
  
  [1X9.4-2 IsVectorInOrbitBySuborbitList[101X
  
  [33X[1;0Y[29X[2XIsVectorInOrbitBySuborbitList[102X( [3Xv[103X, [3Xobsol[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xfail[109X or an integer[133X
  
  [33X[0;0YChecks  probabilistically,  if  the  element  [3Xv[103X lies in one of the partially
  enumerated orbit-by-suborbits in the orbit-by-suborbit list object [3Xobsol[103X. If
  yes,  the  number  of  that  orbit-by-suborbit is returned and the answer is
  guaranteed to be correct. If the answer is [9Xfail[109X there is a small probability
  that  the  point  actually  lies  in one of the orbits but this could not be
  shown.[133X
  
  [1X9.4-3 OrbitsFromSeedsToOrbitList[101X
  
  [33X[1;0Y[29X[2XOrbitsFromSeedsToOrbitList[102X( [3Xobsol[103X, [3Xli[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YTakes  the elements in the list [3Xli[103X as seeds for orbit-by-suborbits. For each
  such   seed   it   is   first   checked  whether  it  lies  in  one  of  the
  orbit-by-suborbits in [3Xobsol[103X, which must be an orbit-by-suborbit list object.
  If  not  found,  51%  of the orbit-by-suborbit of the seed is enumerated and
  added to the list [3Xobsol[103X.[133X
  
  [33X[0;0YThis  function  is  a  good  way  to  quickly  enumerate a greater number of
  orbit-by-suborbits.[133X
  
  [1X9.4-4 VerifyDisjointness[101X
  
  [33X[1;0Y[29X[2XVerifyDisjointness[102X( [3Xobsol[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThis  function  checks  deterministically, whether the orbit-by-suborbits in
  the  orbit-by-suborbit list object [3Xobsol[103X are disjoint or not and returns the
  corresponding  boolean  value.  This  is not a Monte-Carlo algorithm. If the
  answer  is  [9Xfalse[109X,  the  function  writes  out,  which  orbits  are  in fact
  identical.[133X
  
  [1X9.4-5 Memory[101X
  
  [33X[1;0Y[29X[2XMemory[102X( [3Xobsol[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns   the   total   memory   used  for  all  orbit-by-suborbits  in  the
  orbit-by-suborbit-list  [3Xobsol[103X.  Precomputation data is not included, ask the
  setup object instead.[133X
  
  [1X9.4-6 TotalLength[101X
  
  [33X[1;0Y[29X[2XTotalLength[102X( [3Xobsol[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns  the  total number of points stored in all orbit-by-suborbits in the
  orbit-by-suborbit-list [3Xobsol[103X.[133X
  
  [1X9.4-7 Size[101X
  
  [33X[1;0Y[29X[2XSize[102X( [3Xobsol[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the total number of points in the orbit-by-suborbit-list [3Xobsol[103X.[133X
  
  [1X9.4-8 SavingFactor[101X
  
  [33X[1;0Y[29X[2XSavingFactor[102X( [3Xobsol[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns   the  quotient  of  the  total  number  of  points  stored  in  all
  orbit-by-suborbits  in the orbit-by-suborbit-list [3Xobsol[103X and the total number
  of  [22XU[122X-minimal  points stored, which is the average saving factor considering
  all   orbit-by-suborbits   together.   Note   that   the   memory   for  the
  precomputations is not considered here![133X
  
