pkg://kash3-2006.01.31.tbz:17419571/
share/
kash3/
gap/
grp/cryst.grp
downloads
#############################################################################
##
#A cryst.grp GAP group library Volkmar Felsch
##
#H @(#)$Id: cryst.grp,v 1.3 1997/03/27 17:50:31 werner Exp $
##
#Y Copyright (C) 1994, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains functions that allow to access most of the data which
## are listed in the tables of the book "Crystallographic groups of four-
## dimensional space" by Harold Brown, Rolf Buelow, Joachim Neubueser, Hans
## Wondratschek, and Hans Zassenhaus (Wiley, New York, 1978).
##
##
## For each of the dimensions 2, 3, and 4, the tables of the book are
## arranged in the following hierarchical format:
## dimension,
## crystal family,
## crystal system,
## Q-class (geometric crystal class),
## Z-class (arithmetic crystal class),
## space-group type.
##
## The following conventions for local variables are used throughout in all
## functions of this library.
##
## dim = dimension,
## sys = crystal system number with respect to a given dimension,
## qcl = Q-class number with respect to given dimension and crystal system,
## zcl = Z-class number with respect to given dimension, crystal system, and
## Q-class,
## sgt = space-group type with respect to given dimension, crystal system,
## Q-class, and Z-class,
## q = Q-class number with respect to the list of all Q-classes of the
## current dimension,
## z = Z-class number with respect to the list of all Z-classes of the
## current dimension,
## t = space-group type with respect to the list of all space-group types
## of the current dimension,
## CR = catalogue record CrystGroupsCatalogue[dim] for the current
## dimension dim.
##
## For most of the functions in this library there are two versions given,
## a public version and an internal version which are distinguished by the
## prefix "CR_" in the name of the internal version and by different
## parameter lists. The reason for that distinction is that in the public
## functions the arguments are checked for being in range whereas no
## argument checking is done in the internal functions.
##
#H $Log: cryst.grp,v $
#H Revision 1.3 1997/03/27 17:50:31 werner
#H Fixed typo in crTransposedSpaceGroupTyp.
#H
#H Revision 1.2 1997/01/14 15:19:24 gap
#H vfelsch added the group size to the group returned by TransposedSpaceGroup and
#H changed AgGroupQClass to return PAG systems
#H
#H Revision 1.1.1.1 1996/12/11 12:37:16 werner
#H Preparing 3.4.4 for release
#H
#H Revision 3.2.1.2 1995/11/28 09:45:39 vfelsch
#H corrected, improved, and extended
#H
#H Revision 3.2.1.1 1995/02/14 08:53:13 vfelsch
#H fixed bug in SpaceGroup
#H
#H Revision 3.2 1994/06/24 08:17:53 vfelsch
#H applied some minor changes
#H
#H Revision 3.1 1994/06/10 16:41:27 mschoene
#H initial revision under RCS
#H
##
#############################################################################
##
#F GLZOps( <dim>, <system>, <qclass> ) . . . . . . . operations for GL(n,Z)
##
## 'GLZOps'
##
GLZOps := Copy( MatGroupOps );
#############################################################################
##
#F GLZOps.\in( <obj>, <G> ) . . . . . . . . . . membership test for GL(n,Z)
##
GLZOps.\in := function( g, G )
return IsMat( g )
and Length( g ) = Length( G.generators[1] )
and ForAll( Flat( g ), IsInt )
and AbsInt( DeterminantMat( g ) ) = 1;
end;
#############################################################################
##
#F GLZOps.Normalizer( <G>, <H> ) . . . . . . . . . . normalizer in GL(n,Z)
##
GLZOps.Normalizer := function ( G, H )
local con, N, param;
if IsBound( H.crZClass ) then
param := H.crZClass;
N := CR_NormalizerZClass( param );
if IsBound( H.crConjugator and H.crConjugator <> H.identity ) then
con := H.crConjugator;
N.generators := con^-1 * N.generators * con;
if IsBound( N.crConjugator ) then
con := N.crConjugator * con;
fi;
N.crConjugator := con;
fi;
return N;
fi;
return MatGroupOps.Normalizer( G, H );
end;
#############################################################################
##
## GLZ2 . . . . . . . . . . . . . . . . . . . general linear group GL(2,Z)
##
GLZ2 := Group(
[ [0,1], [1,0] ],
[ [1,0], [1,1], ],
[ [-1,0], [0,1], ] );
GLZ2.operations := GLZOps;
GLZ2.name := "GLZ2";
GLZ2.size := "infinity";
#############################################################################
##
## GLZ3 . . . . . . . . . . . . . . . . . . . general linear group GL(3,Z)
##
GLZ3 := Group(
[ [0,1,0], [0,0,1], [1,0,0] ],
[ [1,0,0], [1,1,0], [0,0,1] ],
[ [-1,0,0], [0,1,0], [0,0,1] ] );
GLZ3.operations := GLZOps;
GLZ3.name := "GLZ3";
GLZ3.size := "infinity";
#############################################################################
##
## GLZ4 . . . . . . . . . . . . . . . . . . . general linear group GL(4,Z)
##
GLZ4 := Group(
[ [0,1,0,0], [0,0,1,0], [0,0,0,1], [1,0,0,0] ],
[ [1,0,0,0], [1,1,0,0], [0,0,1,0], [0,0,0,1] ],
[ [-1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1] ] );
GLZ4.operations := GLZOps;
GLZ4.name := "GLZ4";
GLZ4.size := "infinity";
#############################################################################
##
#F CrystalGroupsCatalogue . . . . . . . . . . . . . . . . . . . . . . . . .
##
## The data listed in this catalogue are essentially the same as those
## listed in the tables of the book "Crystallographic groups of four-
## dimensional space" by Harold Brown, Rolf Buelow, Joachim Neubueser, Hans
## Wondratschek, and Hans Zassenhaus (Wiley, New York, 1978).
##
## For each of the dimensions 2, 3, and 4 the catalogue contains a record
## 'CR', say, which consists of the folowing components.
##
## 'CR.bZClass' is a Boolean list containing the numbers of those Z-classes
## which are Bravais Z-classes.
##
## 'CR.codedConjugatorDadeGroup' is a list of matrices used to conjugate
## Z-class representative groups into subgroups of Dade groups. Each
## entry is of the form p1 + 140 * p2 + 140^2 * p3 + ... + 140^(dim-1)
## * p_dim, where dim is the given dimension, and the pi's are
## pointers to the list 'CR.rowConjugatorDadeGroup'. (It is addressed
## by pointers which are contained in entries of the list
## 'CR.codedDadeGroupsZClass'.)
##
## 'CR.codedDadeGroupsZClass' is a list which for each Z-class contains in
## form of a sublist of consecutive entries representatives of those
## conjugacy classes of subgroups of Dade groups which contain groups
## from that Z-class. Each entry is of the form d or p * 10 + d, where
## d is the number of a Dade group and p is a pointer to the list
## 'CR.codedConjugatorDadeGroup'. (See also list
## 'CR.nullDadeGroupsZClass'.)
##
## 'CR.codedDecompositionQClass' is a list which for each Q-class contains
## an entry of the form t1 or of the form 10 * t2 + t1, where t1 is a
## pointer to the text string list 'CR_TextStrings.QClass' and t2 is a
## pointer to the text string list 'CR_TextStrings.QConstituents'.
##
## 'CR.codedGeneratorsSpaceGroup' is a list which for each space-group type
## that is not the representative of its Z-Class contains an entry of
## the form p1 + m * p2 + m^2 * p3 + ... + m^(n-1) * pn, where m equals
## 'CR.modulSp', and n is the number of the given non-translation
## generators of the space group, and the pi's are pointers to the list
## 'CR.columnGeneratorSpaceGroup'.
##
## 'CR.codedGeneratorZClass' is a list which for each Z-class representative
## matrix group contains its generators in form of a sublist of
## consecutive entries. Each entry is of the form p1 + m * p2 + m^2 * p3
## + ... + m^(dim-1) * p_dim, where m equals 'CR.modulZ', and dim is
## the given dimension, and the pi's are pointers to the list
## 'CR.rowGeneratorZClass'. (See also list 'CR.nullGeneratorsZClass'.)
##
## 'CR.codedIsomorphismTypeQClass' provides for each Q-class an entry of the
## form 100 * t + p, where t is the isomorphism type number of the
## respective representative group (with respect to all groups of the
## same order), and p is either zero or a pointer to the text string
## list 'CR_TextStrings.isomorphismType'.
##
## 'CR.codedNormalizerZClass' provides for each Z-class an entry of the form
## 1000 * n2 + n1, where each of n1 and n2 is either zero or a pointer
## to the list 'CR.GeneratorsZClass'. Let us abbreviate that list by
## "gens". Then the normalizer of the representative matrix group of the
## given Z-class can be computed from n1 and n2 as follows. If n1 = 0,
## then the normalizer is one of the Z-class represenative matrix
## groups, and n2 is a pointer to its generators. If n2 = 0, then the
## normalizer is infinite and will be generated by the given group
## together with the list of generators to which n1 points. If both,
## n1 and n2, are non-zero, then n1 point to the generators of some
## Z-class representative which has to be conjugated by the first
## matrix in the lisin pointed to by nin in order to get the normalizer.
##
## 'CR.codedPresentationQClass' provides for each Q-class an entry of the
## form 10 * p + n, where n is the number of generators of the
## representative group of the given Q-class, and p is a pointer to the
## corresponding list of relators in 'CR.relatorNumbersQClass'.
##
## 'CR.codedPropertiesFamily' provides for each crystal family an entry of
## the form 1000 * f + 100 * b + 10 * q + r, where f is the number of
## free parameters, b is the number of Bravais systems in the family,
## and q and r are pointers to appropriate text strings in the list
## 'CR_TextStrings.family' which describe the rational and the real
## decomposition pattern, respectively.
##
## 'CR.codedPropertiesZClass' provides for each Z-class an entry of the form
## 100 * c + 10 * b + d, where c is the order of the associated
## cohomology group, b is the Bravais type of the given Z-class, and
## d is a pointer to a text string in the list 'CR_TextStrings.ZClass'
## which describes the decomposition pattern.
##
## 'CR.columnGeneratorSpaceGroup' is a list of all different columns which
## occur as last column in at least one of the space group generators
## listed in the catalogue. Note that each column in the list has been
## multiplied by the common denominator of its entries.
##
## 'CR.crystalSystemQClass' provides for each Q-class the number of the
## crystal system to which it belongs.
##
## 'CR.dimension' is the given dimension, i.e. 2, 3, or 4.
##
## 'CR.familyCrystalSystem' provides for each crystal system the family to
## which it belongs.
##
## 'CR.fixedPointFreeSpaceGroup' is a Boolean list containing those
## space-group types for which the corresponding space groups are
## fixed-point-free.
##
## 'CR.GLZ is the group GL(dim,Z), where dim is the given dimension.
##
## 'CR.HermannMauguinSymbol' is a list of the Hermann-Mauguin associated to
## the given dimension in the order in which they are listed in the
## International Tables. (Only for dimensions 2 or 3.)
##
## 'CR.hQClass' is a Boolean list containing the numbers of those Q-classes
## which are holohedries.
##
## 'CR.internatTableSpaceGroupType' provides for each space-group type its
## number in the International Tables. If a space-group type splits into
## an enantiomorphic pair, then the corresponding entry of the list
## contains both numbers in the form 1000 * it1 + it2. (Only for
## dimension 3.)
##
## 'CR.modulSp' is an integer used to pack and unpack the entries of list
## CR.codedGeneratorsSpaceGroup.
##
## 'CR.modulZ is an integer used to pack and unpack the entries of list
## CR.codedGeneratorZClass.
##
## 'CR.nameCrystalFamily' provides for each crystal family its family name.
##
## 'CR.nullDadeGroupsZClass' contains for each Z-class a pointer p to the
## list 'CR.codedDadeGroupsZClass' such that, for some n, the entries
## p + 1 to p + n of that list provide representatives of those
## conjugacy classes of subgroups of Dade groups which contain groups
## from the respective Z-class. The last entry points just to the last
## generator in the list.
##
## 'CR.nullGeneratorsZClass' contains for each Z-class a pointer p to the
## list 'CR.codedGeneratorZClass' such that, for some n, the entries
## p + 1 to p + n of that list provide the generators of the
## representative matrix group of the respective Z-class. Additional
## entries point to sublists which represent generators of infinite
## normalizers or conjugating elements which are used to compute certain
## finite normalizers. The last entry points just to the last generator
## in the list.
##
## 'CR.nullQClass' provides for each crystal system the number of Q-classes
## in the preceding crystal systems. An additional entry gives the
## number of all Q-classes.
##
## 'CR.nullSpaceGroup' provides for each Z-class the number of space-group
## types in the preceding Z-classes. An additional entry gives the
## number of all space-group types.
##
## 'CR.nullZClass' provides for each Q-class the number of Z-classes in the
## preceding Q-classes. An additional entry gives the number of all
## Z-classes.
##
## 'CR.orbitLengthSpaceGroup' provides for each space-group type which is
## not the Z-class representative the orbit length associated with that
## space-group type.
##
## 'CR.orderQClass' provides for each Q-class the order of the groups in
## that Q-class.
##
## 'CR.parametersDadeGroup' is a list which for each Dade group of the given
## dimension contains its parameter list [ dim, sys, qcl, zcl ].
##
## 'CR.QClassZClass' provides for each Z-class the number of the Q-class to
## which it belongs.
##
## 'CR.relatorNumbersQClass' is a list of lists of relators, where the
## relators in the lists are represented by pointers to the list
## 'CR.relatorWordsQClass'.
##
## 'CR.rowConjugatorDadeGroup' is a list of all different rows which occur
## in the matrices which are used to conjugate Z-class representative
## groups into subgroups of Dade groups. (It is addressed by pointers
## which are contained in entries of the list
## 'CR.codedConjugatorDadeGroup'.)
##
## 'CR.rowGeneratorZClass' is a list of all different rows which occur in
## the matrices which are used to generate the Z-class representative
## groups. (It is addressed by pointers which are contained in entries
## of the lists 'CR.nullGeneratorsZClass' and 'CR.codedGeneratorZClass'.
##
## 'CR.spaceGroupIdentity' is the identity matrix of the associated space
## groups.
##
## 'CR.spaceGroupTypeInternatTable' provides for each International Table
## number the corresponding "global parameters" z und t (cf. function
## 'CR_Parameters') in the form 100 * t + z. (Only for dimensions 2 and
## 3.)
##
## 'CR.splittingQClass' is a Boolean list containing the numbers of those
## Q-classes which split into enantiomorphic pairs. (Only for dimension
## 4.)
##
## 'CR.splittingSpaceGroupType' is a Boolean list containing those
## space-group types which split into enantiomorphic pairs. (Only for
## dimensions 3 and 4.)
##
## 'CR.splittingZClass' is a Boolean list containing the numbers of those
## Z-classes which split into enantiomorphic pairs. (Only for dimension
## 4.)
##
## 'CR_TextStrings.roman' is a list of the 23 smallest non-negative integers
## expressed by Roman numerals.
##
CR_TextStrings := rec( );
CR_2 := rec( );
CR_3 := rec( );
CR_4 := rec( );
CrystGroupsCatalogue := [ CR_TextStrings, CR_2, CR_3, CR_4 ];
CR_TextStrings.crystalSystem :=
["#I Crystal system "," Q-class"," Q-classes","; holohedry (",",",": ",
")"];
CR_TextStrings.family :=
["irreducible","decomposition pattern ","1+1","1+1+1","1+1+1+1",
"1+1+2","1+2","1+3","2+2","Q-","R-","#I Family ",": ","; ","s",
" free parameter"," crystal system"," Bravais flock",";\n#I "];
CR_TextStrings.isomorphismType :=
["C1","C2","C3","C2xC2","C4","C5","C6","D6","C2xC2xC2","C4xC2",
"C8","D8","Q8","C3xC3","C10","D10","C6xC2","C12","D12","Q12",
"A4","C2xC2xC2xC2","C4xC2xC2","C4xC4","D8xC2","<2,2,2>2","(4,4/2,2)",
"<2,2/4;2>","<2,2/2>","D16","<-2,4/2>","C6xC3","D6xC3","D(C3xC3)",
"D10xC2","C6xC2xC2","C12xC2","D8xC3","Q8xC3","D12xC2","D6xC4",
"Q12xC2","<-2,2,3>","A4xC2","(4,6/2,2)","D24","<2,3,3>","S4",
"D8xC2xC2","D8xC4","C4^S2","(C2xC2)^S2","D(C4xC4)","C6xC6","D6xC6",
"Q12xC3","D(C3xC3)xC2","D6xD6","(3,4,4;3)","D8xC6","D12xC2xC2",
"D12xC4","A4xC2xC2","(4,6/2,2)xC2","D24xC2","S4xC2","D8xD6","A5",
"D8xD8","D12xC6","D(C6xC6)","(4,6/2,2)xC3","<2,3,3>xC3","D12xD6",
"(3,4,4;3)xC2","D6^S2","S4xC2xC2","D12xD8","S5","A5xC2","(D6^S2)xC2",
"D12xD12","S5xC2","D12^S2","C2^S4","[3,4,3]"];
CR_TextStrings.QClass :=
["Q-constituents ","Q-irreducible; ","R-irreducible; ","C-irreducible; ",
"; cc; ","; ncc; ","#I Q-class ","#I *Q-class ","): size ",
"; isomorphism type "," Z-class"," Z-classes"," space group",
" space grps",",","(",".","H ","; "," = ",";\n#I "];
CR_TextStrings.QConstituents :=
["(2,1,1)+(2,1,2)","(4,1,1)+(4,1,2)+(4,3,1)","(4,1,1)+(4,1,2)+(4,3,2)",
"(4,1,1)+(4,1,2)+(4,4,1)","(4,1,1)+(4,1,2)+(4,4,2)",
"(4,1,1)+(4,1,2)+(4,4,3)","(4,1,1)+(4,1,2)+(4,4,4)","(3,1,1)+2*(3,1,2)",
"(4,1,1)+3*(4,1,2)","(3,1,1)+(3,3,1)","(3,1,1)+(3,3,2)","(3,1,1)+(3,4,1)",
"(3,1,1)+(3,4,2)","(3,1,1)+(3,4,3)","(3,1,1)+(3,4,4)","(4,1,1)+(4,7,1)",
"(4,1,1)+(4,7,2)","(4,1,1)+(4,7,3)","(4,1,1)+(4,7,4)","(4,1,1)+(4,7,5)",
"(3,1,2)+(3,3,1)","(3,1,2)+(3,3,2)","(3,1,2)+(3,4,1)","(3,1,2)+(3,4,2)",
"(3,1,2)+(3,4,3)","(3,1,2)+(3,4,4)","(4,1,2)+(4,7,1)","(4,1,2)+(4,7,2)",
"(4,1,2)+(4,7,3)","(4,1,2)+(4,7,4)","(4,1,2)+(4,7,5)","2*(2,1,1)",
"2*(3,1,1)+(3,1,2)","2*(4,1,1)+2*(4,1,2)","2*(4,1,1)+(4,3,1)",
"2*(4,1,1)+(4,3,2)","2*(4,1,1)+(4,4,1)","2*(4,1,1)+(4,4,2)",
"2*(4,1,1)+(4,4,3)","2*(4,1,1)+(4,4,4)","2*(2,1,2)","2*(4,1,2)+(4,3,1)",
"2*(4,1,2)+(4,3,2)","2*(4,1,2)+(4,4,1)","2*(4,1,2)+(4,4,2)",
"2*(4,1,2)+(4,4,3)","2*(4,1,2)+(4,4,4)","2*(4,3,1)","2*(4,3,2)",
"2*(4,4,1)","2*(4,4,2)","2*(4,4,3)","2*(4,4,4)","3*(3,1,1)",
"3*(4,1,1)+(4,1,2)","3*(3,1,2)","(4,3,1)+(4,3,2)","(4,3,1)+(4,4,1)",
"(4,3,1)+(4,4,2)","(4,3,1)+(4,4,3)","(4,3,1)+(4,4,4)","(4,3,2)+(4,4,1)",
"(4,3,2)+(4,4,2)","(4,3,2)+(4,4,3)","(4,3,2)+(4,4,4)","4*(4,1,1)",
"4*(4,1,2)","(4,4,1)+(4,4,2)","(4,4,1)+(4,4,3)","(4,4,1)+(4,4,4)",
"(4,4,2)+(4,4,3)","(4,4,2)+(4,4,4)","(4,4,3)+(4,4,4)"];
CR_TextStrings.spaceGroup :=
["#I Space-group type ","#I *Space-group type ","; IT(",
"; fp-free","orbit size ","(",",",") = ",", IT(","; ",";\n#I ",
")","#I The non-translation generators of "," are\n\n"];
CR_TextStrings.ZClass :=
["; Z-irreducible","; Z-reducible","; Z-decomposable",
"; fully Z-reducible","#I Z-class ","#I *Z-class ",
": Bravais type "," space group"," space groups",
"; cohomology group size "," = Z(",")",",","(","B ","/",";\n#I "];
CR_TextStrings.roman :=
["I","II","III","IV","V","VI","VII","VIII","IX","X","XI","XII","XIII",
"XIV","XV","XVI","XVII","XVIII","XIX","XX","XXI","XXII","XXIII"];
CR_2.bZClass := BlistList( [1..13], [2,5,6,8,13] );
CR_2.codedConjugatorDadeGroup := [142,421,563];
CR_2.codedDadeGroupsZClass := [2,1,2,1,21,12,32,21,1,12,1,1,1,2,2,2,2,2];
CR_2.codedDecompositionQClass := [325,415,16,416,3,4,3,4,3,4];
CR_2.codedGeneratorsSpaceGroup := [2,1,3,3];
CR_2.codedGeneratorZClass :=
[16,6,10,26,10,6,26,6,25,6,10,25,6,19,1,19,26,19,6,19,26,6,19,10,2,17];
CR_2.codedIsomorphismTypeQClass := [101,102,102,104,205,412,103,208,107,319];
CR_2.codedNormalizerZClass := [14000,14000,5,6,8,8,8,8,13,13,13,13,13];
CR_2.codedPresentationQClass := [11,21,21,202,212,243,401,462,452,433];
CR_2.codedPropertiesFamily := [3133,2233,1101,1101];
CR_2.codedPropertiesZClass :=
[114,114,214,122,414,122,111,211,111,111,111,111,111];
CR_2.columnGeneratorSpaceGroup := [[0,0,1],[0,1,2],[1,0,2],[1,1,2]];
CR_2.crystalSystemQClass := [1,1,2,2,3,3,4,4,4,4];
CR_2.dimension := 2;
CR_2.familyCrystalSystem := [1..4];
CR_2.fixedPointFreeSpaceGroup := BlistList( [1..17], [1,4] );
CR_2.GLZ := GLZ2;
CR_2.HermannMauguinSymbol :=
["p1","p2","pm","pg","cm","p2mm","p2mg","p2gg","c2mm","p4","p4mm",
"p4gm","p3","p3m1","p31m","p6","p6mm"];
CR_2.hQClass := BlistList( [1..10], [2,4,6,10] );
CR_2.modulSp := 4;
CR_2.modulZ := 6;
CR_2.nameCrystalFamily := ["oblique","rectangular","square","hexagonal"];
CR_2.nullDadeGroupsZClass := [0,2,4,5,8,9,11,12,13,14,15,16,17,18];
CR_2.nullGeneratorsZClass := [0,1,2,3,4,6,8,10,13,14,16,18,20,23,26];
CR_2.nullQClass := [0,2,4,6,10];
CR_2.nullSpaceGroup := [0,1,2,4,5,8,9,10,12,13,14,15,16,17];
CR_2.nullZClass := [0,1,2,4,6,7,8,9,11,12,13];
CR_2.orbitLengthSpaceGroup := [1,2,1,1];
CR_2.orderQClass := [1,2,2,4,4,8,3,6,6,12];
CR_2.parametersDadeGroup := [[2,3,2,1,0],[2,4,4,1,0]];
CR_2.QClassZClass := [1,2,3,3,4,4,5,6,7,8,8,9,10];
CR_2.relatorNumbersQClass :=
[[1], [2], [2,10,22,28,40,61,92,129,163,198],
[2,10,22,31,34,40,61,92,116,129,166,186,199,228,247],
[2,10,22,31,40,61,92,129,174,202], [2,10,22,40,61,129],
[2,10,24,28,34,40,62,92,116,142,168,186,201,227,249],
[2,10,24,28,40,62,92,142,163,198], [2,10,24,28,40,62,92,142,168,201],
[2,10,24,28,40,62,92,142,168,214],
[2,10,24,30,40,78,80,100,106,108,109,130,153,175,176,177,179,210,217,219,
220],
[2,10,24,33,40,78,80,100,107,108,110,111,112,113,114,115,130,153,175,176,
178,180,181,182,183,184,185,211,218,219,221,222,223,224,225,226],
[2,10,24,40,62,142], [2,10,26,28,40,62,104,142,163,209],
[2,10,26,28,40,78,92,130,168,209], [2,10,26,28,40,78,92,130,168,216],
[2,10,26,28,40,78,93,130,164,216], [2,10,26,40,62,142],
[2,10,26,40,78,130], [2,10,40], [2,11,40],
[2,11,22,28,40,61,92,142,163,198], [2,11,22,40,61,129],
[2,11,22,40,61,142],
[2,17,22,28,35,38,40,61,105,117,125,147,172,189,195,198,230,237,244,250,
256],
[2,17,22,28,36,37,40,61,105,120,124,147,172,190,194,198,231,236,241,251,
259],
[2,17,22,28,36,40,61,105,118,147,172,191,198,233,243],
[2,17,22,32,34,40,79,99,116,129,172,192,205,228,248],
[2,17,22,32,35,37,39,40,79,96,122,123,128,147,167,186,196,197,205,234,239,
240,245,252,255,260,261,262],
[2,17,22,32,35,37,40,79,96,122,123,147,167,186,196,205,234,239,245,252,
260],
[2,17,22,32,35,40,79,98,121,147,165,188,212,232,246],
[2,17,22,32,40,79,95,129,172,199], [2,17,25,28,40,79,92,133,168,215],
[2,17,25,40,79,133], [2,17,26,28,40,61,92,147,163,216],
[2,19,26,28,40,61,92,136,173,216], [2,19,26,28,40,61,92,148,173,216],
[2,19,26,40,61,136], [2,19,26,40,61,148], [3],
[3,10,22,28,40,73,92,129,163,198], [3,10,22,40,61,129],
[3,10,22,40,73,129], [3,10,22,41,61,129], [3,10,40], [3,10,41],
[3,16,22,40,73,146], [3,16,25,28,40,66,101,143,169,215],
[3,16,25,40,79,131], [3,16,40], [3,19,22,41,61,148], [3,19,22,41,73,137],
[3,19,25,28,40,79,101,137,168,207], [3,19,25,28,40,79,105,137,173,215],
[3,19,25,40,79,137], [3,19,41], [3,21,22,41,61,149], [3,21,22,41,73,136],
[3,21,41], [4],
[4,10,22,28,35,37,40,74,102,119,126,129,169,187,193,198,229,238,242,253,
257],
[4,10,22,28,35,37,40,74,102,119,127,129,169,187,193,198,229,235,242,254,
258],
[4,10,22,28,35,40,74,102,119,129,169,187,198,229,242],
[4,10,22,28,40,63,102,129,163,200], [4,10,22,28,40,74,92,129,163,198],
[4,10,22,28,40,74,102,129,169,198], [4,10,22,40,61,129],
[4,10,22,40,63,129], [4,10,22,40,74,129], [4,10,22,40,74,143],
[4,10,22,40,78,129], [4,10,22,42,61,129], [4,10,40], [4,10,42],
[4,12,24,28,42,63,102,130,170,202], [4,12,24,28,42,78,102,132,168,214],
[4,12,24,29,42,63,92,130,168,203], [4,12,24,42,63,130],
[4,12,24,42,78,132], [4,12,27,28,42,62,102,144,170,208],
[4,12,27,42,63,132], [4,12,42], [4,17,22,28,40,79,102,147,172,213],
[4,17,22,40,67,129], [4,17,22,40,74,147], [4,17,22,42,74,147],
[4,17,25,28,40,62,102,132,163,206], [4,17,25,40,67,144], [4,17,40],
[4,17,42], [4,18,26,48,50,52,54,71,82,84,87,89,139,150,154,157,160],
[4,18,26,48,50,52,54,72,81,85,86,90,140,151,156,159,161],
[4,20,26,49,51,53,55,56,57,58,59,60,70,83,84,88,91,141,152,155,158,162],
[5], [5,10,43], [5,17,42], [6], [6,10,22,28,44,61,92,129,163,198],
[6,10,22,40,61,129], [6,10,22,40,62,132], [6,10,22,40,64,129],
[6,10,22,40,75,129], [6,10,22,40,75,144], [6,10,22,44,61,129],
[6,10,22,44,75,145], [6,10,40], [6,10,44], [6,13,44], [6,16,22,40,75,146],
[6,16,40], [6,17,22,44,75,134], [6,17,44], [6,19,22,40,75,148],
[6,19,22,44,61,148], [6,19,22,44,75,137], [6,19,22,44,75,148],
[6,19,25,28,40,69,92,138,173,215], [6,19,25,40,68,135], [6,19,40],
[6,19,44], [7], [7,10,22,28,44,64,97,129,163,212],
[7,10,22,28,44,76,94,129,163,212], [7,10,22,44,64,129],
[7,10,22,44,65,145], [7,10,22,44,76,129], [7,10,42], [7,10,44], [7,10,45],
[8], [8,10,46], [8,17,42], [9], [9,10,22,40,76,129], [9,10,22,40,77,129],
[9,10,22,44,61,129], [9,10,22,44,76,129], [9,10,22,44,77,145],
[9,10,23,47,76,144], [9,10,40], [9,10,44], [9,10,45], [9,10,47],
[9,14,24,28,45,65,103,132,171,204], [9,14,24,45,65,132], [9,14,45],
[9,15,44]];
CR_2.rowConjugatorDadeGroup := [[0,-1],[1,-1],[1,0],[1,1]];
CR_2.rowGeneratorZClass := [[-1,0],[0,-1],[0,1],[1,-1],[1,0],[1,1]];
CR_2.spaceGroupIdentity :=
[[1,0,0],[0,1,0],[0,0,1]];
CR_2.spaceGroupTypeInternatTable :=
[101,202,303,403,504,605,705,805,906,1007,1108,1208,1309,1410,
1511,1612,1713];
CR_3.bZClass :=
BlistList( [1..73], [2,7,8,18,19,20,21,36,37,48,58,71,72,73] );
CR_3.codedConjugatorDadeGroup :=
[79395,118877,176683,176831,177111,177382,177535,178513,178786,
178792,196429,196575,217297,217709,256914,294149,296253,296396,
334898];
CR_3.codedDadeGroupsZClass :=
[1,2,3,4,1,2,3,4,141,142,191,51,142,123,23,144,134,141,42,
31,81,42,93,13,174,74,141,142,191,81,142,93,123,174,134,2,
61,2,3,184,13,4,2,61,2,51,61,142,143,184,104,113,13,144,
2,61,2,3,184,13,4,2,13,144,2,13,144,2,13,144,142,13,144,
2,13,144,142,2,13,144,13,144,142,13,144,42,163,154,1,42,
163,154,1,42,163,154,1,1,42,163,154,1,1,42,163,154,1,1,1,
1,1,1,1,1,1,1,2,3,4,2,3,4,2,3,4,2,3,4,2,3,4];
CR_3.codedDecompositionQClass :=
[545,565,85,335,565,566,86,566,106,216,216,226,116,226,226,
126,256,246,136,266,146,236,256,266,156,246,266,4,4,4,4,4];
CR_3.codedGeneratorsSpaceGroup :=
[4,1,1,14,56,70,14,84,88,102,14,112,56,1,57,62,63,14,70,
71,76,1,14,112,1,113,18,112,8,14,798,1666,1176,1190,1568,
1582,1232,1246,1624,1638,112,126,1428,1442,14,196,1568,1582,
1764,252,112,1568,1092,17,1,104,14,1624,1638,994,238,14,1656,
1922,1670,994,14,1,15,1658,1672,1659,1673,8,994,1002,14,1656,
1670,1,1658,1659,8,994,14,196,210,23212,23226,23408,23422,
23184,23198,23380,23394,112,126,308,322,112,20090,20062,2,28,
28,113,1,1,1582,14,14,28,1,29,14,392,14,406,1,14,15,1,14,
14,196,210,1092,1092,23324,15288,3528,15288,30156,23436,23436,
20194,23325,3536,20194,1582,326550,326648,126,49504,49406,282716];
CR_3.codedGeneratorZClass :=
[3092,2624,2719,2570,2997,3164,2624,2719,2624,2570,2719,2985,
2570,2985,5062,276,4433,2783,2997,2985,3164,2985,2931,2570,
1071,276,1643,2783,2624,2719,2985,2624,2570,2985,2624,5062,276,
2624,4433,2783,3159,2985,1625,2783,2575,2985,4451,2783,2624,
3159,2985,2624,1625,2783,2719,3159,2985,4433,1625,2783,2997,
3159,2985,1643,1625,2783,2719,2575,2985,2997,2575,2985,1643,
4451,2783,4433,4451,2783,2624,2719,3159,2985,2624,4433,1625,
2783,3884,3121,2624,3884,2624,3121,2570,3884,2570,3121,2803,
3121,3164,3884,2931,3121,3164,3121,2624,2570,3884,2624,2570,
3121,2624,2803,3121,2985,3121,2731,3121,2624,2985,3121,2803,
2985,3121,2931,2985,3121,2931,2731,3121,2803,2731,3121,2624,
2803,2985,3121,3884,2719,2985,3884,5062,276,3884,4433,2783,
2624,3884,2719,2985,2624,3884,5062,276,2624,3884,4433,2783,
2803,3884,2719,2985,6595,3884,5062,276,3005,3884,4433,2783,
2931,3884,2719,2985,204,3884,5062,276,2711,3884,4433,2783,2624,
2803,3884,2719,2985,2624,6595,3884,5062,276,2624,3005,3884,
4433,2783,979,3080,3884,3095,5252,3093,920,824,2636,3884,3095,
5980,3112,424,458,3093,824,2636,3093,5157,2985,2719,3112,458,
2783,3112,6052,3005,2570];
CR_3.codedIsomorphismTypeQClass :=
[101,102,102,102,104,104,104,109,205,205,210,412,412,412,625,
103,107,208,208,319,107,107,117,319,319,319,640,521,1044,
1548,1548,3666];
CR_3.codedNormalizerZClass :=
[75000,77000,76000,78000,80000,82000,79000,81000,71,36,72,73,36,
36,19,74037,37,71,36,72,73,36,37,36,37,36,37,36,37,36,37,36,36,
37,37,36,37,48,58,48,58,48,58,58,48,58,58,48,58,58,58,58,58,58,
58,58,58,58,71,72,73,71,72,73,71,72,73,71,72,73,71,72,73];
CR_3.codedPresentationQClass :=
[11,21,21,21,202,202,202,63,212,212,233,243,243,243,224,401,
452,462,462,443,452,452,423,433,433,433,414,133,84,94,94,75];
CR_3.codedPropertiesFamily := [6144,4244,3444,2277,2277,1301];
CR_3.codedPropertiesZClass :=
[114,114,214,123,414,223,814,223,814,223,132,242,1614,423,423,
232,442,6414,823,232,842,414,222,114,122,414,222,814,222,814,
422,414,414,222,222,1614,422,112,324,112,124,112,324,324,212,
224,224,212,224,224,624,124,224,624,424,224,224,424,211,121,
231,411,221,231,411,221,231,211,221,231,411,421,231];
CR_3.columnGeneratorSpaceGroup :=
[[0,0,0,1],[0,0,1,2],[0,0,1,3],[0,0,1,4],[0,1,0,2],[0,1,1,2],
[1,0,0,2],[1,0,1,2],[1,1,0,2],[1,1,1,2],[1,1,2,4],[2,0,1,4],
[3,1,0,4],[3,2,3,4]];
CR_3.crystalSystemQClass :=
[1,1,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7];
CR_3.dimension := 3;
CR_3.familyCrystalSystem := [1,2,3,4,5,5,6];
CR_3.fixedPointFreeSpaceGroup :=
BlistList( [1..219], [1,4,7,9,19,33,34,76,142,165] );
CR_3.GLZ := GLZ3;
CR_3.HermannMauguinSymbol :=
["P1","P-1","P2","P21","C2","Pm","Pc","Cm","Cc","P2/m","P21/m",
"C2/m","P2/c","P21/c","C2/c","P222","P2221","P21212","P212121",
"C2221","C222","F222","I222","I212121","Pmm2","Pmc21","Pcc2",
"Pma2","Pca21","Pnc2","Pmn21","Pba2","Pna21","Pnn2","Cmm2","Cmc21",
"Ccc2","Amm2","Abm2","Ama2","Aba2","Fmm2","Fdd2","Imm2","Iba2",
"Ima2","Pmmm","Pnnn","Pccm","Pban","Pmma","Pnna","Pmna","Pcca",
"Pbam","Pccn","Pbcm","Pnnm","Pmmn","Pbcn","Pbca","Pnma","Cmcm",
"Cmca","Cmmm","Cccm","Cmma","Ccca","Fmmm","Fddd","Immm","Ibam",
"Ibca","Imma","P4","P41","P42","P43","I4","I41","P-4","I-4",
"P4/m","P42/m","P4/n","P42/n","I4/m","I41/a","P422","P4212",
"P4122","P41212","P4222","P42212","P4322","P43212","I422","I4122",
"P4mm","P4bm","P42cm","P42nm","P4cc","P4nc","P42mc","P42bc","I4mm",
"I4cm","I41md","I41cd","P-42m","P-42c","P-421m","P-421c","P-4m2",
"P-4c2","P-4b2","P-4n2","I-4m2","I-4c2","I-42m","I-42d","P4/mmm",
"P4/mcc","P4/nbm","P4/nnc","P4/mbm","P4/mnc","P4/nmm","P4/ncc",
"P42/mmc","P42/mcm","P42/nbc","P42/nnm","P42/mbc","P42/mnm",
"P42/nmc","P42/ncm","I4/mmm","I4/mcm","I41/amd","I41/acd","P3",
"P31","P32","R3","P-3","R-3","P312","P321","P3112","P3121",
"P3212","P3221","R32","P3m1","P31m","P3c1","P31c","R3m","R3c",
"P-31m","P-31c","P-3m1","P-3c1","R-3m","R-3c","P6","P61","P65",
"P62","P64","P63","P-6","P6/m","P63/m","P622","P6122","P6522",
"P6222","P6422","P6322","P6mm","P6cc","P63cm","P63mc","P-6m2",
"P-6c2","P-62m","P-62c","P6/mmm","P6/mcc","P63/mcm","P63/mmc","P23",
"F23","I23","P213","I213","Pm-3","Pn-3","Fm-3","Fd-3","Im-3",
"Pa-3","Ia-3","P432","P4232","F432","F4132","I432","P4332","P4132",
"I4132","P-43m","F-43m","I-43m","P-43n","F-43c","I-43d","Pm-3m",
"Pn-3n","Pm-3n","Pn-3m","Fm-3m","Fm-3c","Fd-3m","Fd-3c","Im-3m",
"Ia-3d"];
CR_3.hQClass := BlistList( [1..32], [2,5,8,15,20,27,32] );
CR_3.internatTableSpaceGroupType :=
[1,2,3,4,5,6,7,8,9,10,13,11,14,12,15,16,17,18,19,21,20,
22,23,24,25,28,27,30,32,34,26,31,29,33,35,37,36,38,39,40,
41,42,43,44,46,45,47,49,50,48,51,54,53,52,57,60,59,56,55,
58,62,61,65,66,63,67,68,64,69,70,71,72,74,73,75,78076,77,
79,80,81,82,83,84,85,86,87,88,89,95091,93,90,96092,94,97,
98,99,105,103,101,100,106,104,102,107,108,109,110,111,112,
113,114,115,116,117,118,119,120,121,122,123,124,131,132,125,
126,133,134,129,130,137,138,127,128,135,136,139,140,141,142,
146,143,145144,148,147,155,149,153151,150,154152,160,161,156,
158,157,159,166,167,162,163,164,165,168,171172,173,170169,174,
175,176,177,180181,182,179178,183,184,186,185,187,188,189,190,
191,192,194,193,195,198,196,197,199,200,201,205,202,203,204,
206,207,213212,208,209,210,211,214,215,218,216,219,217,220,
221,223,222,224,225,226,228,227,229,230];
CR_3.modulSp := 14;
CR_3.modulZ := 19;
CR_3.nameCrystalFamily :=
["triclinic","monoclinic","orthorhombic","tetragonal","hexagonal",
"cubic"];
CR_3.nullDadeGroupsZClass :=
[0,4,8,10,17,19,26,28,35,36,38,40,42,43,45,48,51,54,55,
57,59,61,62,64,65,67,68,70,71,73,74,76,77,78,80,82,83,85,
88,89,92,93,96,97,98,101,102,103,106,107,108,109,110,111,
112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,
127,128,129,130,131];
CR_3.nullGeneratorsZClass :=
[0,1,2,3,4,5,6,8,10,12,14,16,18,20,22,24,26,28,31,34,37,
40,42,44,46,48,51,54,57,60,63,66,69,72,75,78,82,86,87,88,
90,92,94,96,98,100,102,104,107,110,113,115,117,120,123,126,
129,132,136,139,142,145,149,153,157,161,165,169,173,177,181,
186,191,196,197,200,204,207,211,214,218,221,225];
CR_3.nullQClass := [0,2,5,8,15,20,27,32];
CR_3.nullSpaceGroup :=
[0,1,2,4,5,7,9,13,15,19,21,22,24,34,37,41,43,46,62,68,
70,74,77,79,80,81,85,87,93,95,103,107,111,115,117,119,135,
139,140,142,143,144,145,147,149,151,153,155,157,159,161,165,
166,168,172,176,178,180,184,186,187,189,192,194,196,199,201,
203,205,207,209,213,217,219];
CR_3.nullZClass :=
[0,1,2,4,6,8,12,17,21,23,25,27,29,31,35,37,39,41,44,47,
50,51,52,53,54,55,57,58,61,64,67,70,73];
CR_3.orbitLengthSpaceGroup :=
[1,3,1,3,1,3,1,3,3,1,1,1,2,1,2,1,1,2,2,2,2,1,2,1,1,
1,1,2,1,3,3,1,6,6,6,6,6,6,3,3,3,3,6,2,1,2,1,1,2,1,
3,3,1,2,1,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,2,2,2,1,1,1,1,1,1,2,1,2,1,2,1,2,1,1,1,1,1,1,1,
1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1];
CR_3.orderQClass :=
[1,2,2,2,4,4,4,8,4,4,8,8,8,8,16,3,6,6,6,12,6,6,12,12,12,
12,24,12,24,24,24,48];
CR_3.parametersDadeGroup :=
[[3,6,7,1,0],[3,7,5,1,0],[3,7,5,2,0],[3,7,5,3,0]];
CR_3.QClassZClass :=
[1,2,3,3,4,4,5,5,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9,10,
10,11,11,12,12,13,13,14,14,14,14,15,15,16,16,17,17,18,18,
18,19,19,19,20,20,20,21,22,23,24,25,26,26,27,28,28,28,29,
29,29,30,30,30,31,31,31,32,32,32];
CR_3.relatorNumbersQClass := CR_2.relatorNumbersQClass;
CR_3.rowConjugatorDadeGroup :=
[[-1,-1,-1],[-1,-1,0],[-1,0,0],[-1,0,1],[-1,1,0],
[0,-1,-1],[0,-1,0],[0,-1,1],[0,0,1],[0,1,-1],
[0,1,0],[0,1,1],[1,-1,0],[1,0,-1],[1,0,0],
[1,0,1],[1,1,1],[1,1,2]];
CR_3.rowGeneratorZClass :=
[[-1,-1,-1],[-1,-1,1],[-1,0,0],[-1,0,1],[-1,1,0],[0,-1,0],
[0,-1,1],[0,0,-1],[0,0,1],[0,1,-1],[0,1,0],[0,1,1],[1,-1,0],
[1,0,-1],[1,0,0],[1,0,1],[1,1,-1],[1,1,0],[1,1,1]];
CR_3.spaceGroupIdentity :=
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]];
CR_3.spaceGroupTypeInternatTable :=
[101,202,303,403,504,605,705,806,906,1007,1207,1408,1107,1307,
1508,1609,1709,1809,1909,2110,2010,2211,2312,2412,2513,3113,
2713,2613,3313,2813,3213,2913,3413,3013,3514,3714,3614,3815,
3915,4015,4115,4216,4316,4417,4617,4517,4718,5018,4818,4918,
5118,5418,5318,5218,5918,5818,5518,6018,5718,5618,6218,6118,
6519,6819,6319,6419,6619,6719,6920,7020,7121,7221,7421,7321,
7522,7622,7722,7622,7823,7923,8024,8125,8226,8326,8426,8526,
8627,8727,8828,9128,8928,9228,9028,9328,8928,9228,9429,9529,
9630,10030,9930,10330,9830,10230,9730,10130,10431,10531,10631,
10731,10832,10932,11032,11132,11233,11333,11433,11533,11634,
11734,11835,11935,12036,12136,12436,12536,13236,13336,12836,
12936,12236,12336,12636,12736,13436,13536,13036,13136,13637,
13737,13837,13937,14139,14239,14239,14038,14441,14340,14643,
14844,14743,14944,14743,14944,14542,15246,15447,15346,15547,
15045,15145,15849,15949,16050,16150,15648,15748,16251,16551,
16551,16351,16351,16451,16652,16753,16853,16954,17254,17254,
17054,17054,17154,17355,17455,17655,17555,17756,17856,17957,
18057,18158,18258,18458,18358,18559,18760,18861,18659,18961,
19062,19162,19363,19463,19564,19262,19664,19765,19965,20066,
20166,20267,19865,19865,20367,20468,20669,20870,20568,20769,
20970,21071,21271,21171,21371,21472,21572,21772,21672,21873,21973];
CR_3.splittingSpaceGroupType :=
BlistList( [ 1 .. 219 ], [76,89,92,142,147,149,163,165,170,172,198] );
CR_4.bZClass :=
BlistList( [1..710],
[2,7,8,12,13,14,34,35,36,37,38,39,53,86,87,88,89,90,91,
92,93,108,109,120,130,131,133,173,242,243,244,245,246,295,
296,329,330,331,332,333,337,338,359,378,379,419,428,464,482,
507,563,564,565,566,567,569,574,576,604,621,636,637,691,710] );
CR_4.codedConjugatorDadeGroup :=
[4273681,6400667,8480660,9279532,11195122,12279467,12298451,12357446,
15623922,16613802,19260299,19325973,19338622,21004202,22465651,
25279713,26836042,28656840,31486984,33244423,33246395,36157062,
36697684,38833653,41968382,44304724,44306282,44363772,45130978,
45337662,45343783,45784833,46572214,47083975,47317495,47397622,
48109758,48237718,49885273,50081235,51253736,52059902,52533832,
55437104,55553703,55555115,55574723,55789055,56101272,56102655,
56162732,57492995,57493012,57494114,58023035,58431983,60708505,
60747994,60822328,60823741,60825332,60933786,61155935,61158054,
61216434,61217245,61274523,61389605,61588992,63349460,63376504,
63493248,63508795,63508934,63546728,63548155,63549675,63604695,
63605108,63605942,63663387,63664324,63804066,63804487,63804624,
63804626,63807003,63811766,63862367,63921812,63980875,64018544,
64021624,64137261,64137683,64137684,64137687,64137703,64140233,
64235684,64236099,64354121,64371353,64373855,64470184,64470461,
64470884,64604452,64604453,65725012,66724894,69053908,69054764,
69056855,69066244,69163383,69508195,69527678,69565344,69626658,
69722984,69800681,69843532,71449321,72153765,74193323,77232724,
77585524,78052841,78052983,78052987,78053401,78053406,78053680,
78053683,78053684,78873935,80453739,82383716,83069624,85693922,
88404015,88424440,89811601,90220655,91360562,93731575,93749676,
93990720,93991022,94000098,94049515,94144560,94145297,94167383,
94318062,94402300,94435083,94525188,96497283,96594861,96595283,
96595284,96646684,96709223,96715422,96849822,96889263,96944864,
97003944,97009124,97084602,97087783,97123763,97123802,97124043,
97160044,97165504,97199664,97204424,97223164,97261260,97261266,
97261401,97261683,97261687,97261824,97261826,97264204,97277358,
97283267,97303124,97318514,97398601,97398883,97426038,98124517,
99809001,101220593,102435678,104963063,105118346,105177144,105179776,
105179923,105391650,105491164,105501275,105515550,105531615,
107883357,108140112,110197006,110610142,110668942,110785343,
110950612,112021303,112270943,112962124,112972335,113058164,
113061524,113137486,113193264,113196183,113196222,113369253,
113375115,113376112,113529243,113686904,113726523,113736472,
113884749,113956684,114747111,115518080,116058522,117390613,
118450126,118684357,118686726,118745508,118778275,118783303,
118798855,118823915,118898424,118898523,118900903,118901264,
118901615,118902315,118918123,119074923,119074962,119075763,
119117465,119117591,119195194,119214521,120488549,121194121,
121542872,121544700,121546903,121646337,121662343,121664503,
121664542,121671914,121758723,121761702,121780558,121781453,
121957692,121958521,122114674,124330002,124488183,126679743,
126681704,126682123,126682124,126916222,126918720,126975403,
127074103,127074142,127095115,127152503,127152542,127187664,
127249422,127251923,127251924,127427176,127444143,127446519,
127486992,127487272,127487552,128465703,129424592,129716346,
129719250,129815762,129818103,129835323,129835362,129837703,
129838442,129874523,129874562,129876942,129879322,129995649,
130346222,131229342,132461757,132579328,132902595,133805883,
133931728,134659472,134896493,135145544,135160113,135482363,
135579684,139346101,140793123,140793722,142416874,143007661,
143007801,143008083,143088733,143245818,143301783,143478468,
143478502,143497783,143534066,143536563,143536602,143556902,
143595783,143596242,143598324,143615702,143623386,143691569,
143694068,143729623,143735822,143804284,143811261,143811401,
143811683,143811684,143889365,143906045,143908987,143909266,
143909401,143909407,143909687,143909824,143912169,143951124,
143984729,144043244,144043526,144046460,144046883,144047024,
144047026,144058224,144063553,144871043,145634912,145990618,
146009419,146120862,146136109,146454023,146459944,146460222,
146469882,146548284,146674135,146695115,146799706,147614890,
149198344,149610766,151966145,154716260,155552903,156808724,
156865704,156905324,156921385,157157744,157355695,157371664,
157433401,157433683,157628564,157629260,157629266,157629401,
157629683,157629684,157631644,158747155,160011634,164016851,
165317789,165958163,166020021,167790612,168410783,170524795,
170682584,170682861,170682866,170683006,170683284,170816844,
170820061,170820066,170820206,170823013,170876764,170918475,
170918492,170935564,170937655,170938232,170977292,170995346,
170996875,171035252,171035655,171035795,171035812,171039457,
171055692,171151980,171152020,171152983,171152984,171153261,
171153401,171153406,171153407,171153543,171153683,171153684,
171153826,171154684,171165583,171251266,171251687,171251824,
171255463,171388052,171389733,171389855,171391552,171485003,
171508026,172169252,172341709,172739034,172742693,173328575,
173628193,174092146,174096229,174136122,174190160,174195053,
175429162,176069896,176075689,176289703,176405792,176425795,
176425812,176525064,176662103,176683562,176835995,176836134,
176837248,176838533,176838672,176839093,176854112,176859549,
176860522,176977282,176978682,177049063,179054884,179267833,
181427494,182286904,182286927,182288714,182326550,182484449,
184815843,184893683,185030926,185090286,187628482,189698662,
190557687,190600682,190811390,192160007,193127686,193323518,
193442795,196026785,198446904,200706302,201673422,203352398,
203411195,204027793,204106189,204359435,204437275,207260107,
209678062,209685034,209708584,210904042,212461485,212461611,
214156126,214894765,214907506,215828706,216925333,220802872,
223509220,227338635,229114262,229133722,233222658,234926586,
236132345,236891767,237759067,237812967,241385693,243228437,244256273];
CR_4.codedDadeGroupsZClass :=
[1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1321,2032,233,
2978,1341,2181,3091,4702,4732,2173,4513,3524,3165,3205,1166,
3537,3058,3509,2219,1321,2032,233,1698,1011,3251,3221,4182,
1372,3233,4073,3184,3215,2115,2026,1977,2128,899,3249,1321,
2032,233,1698,1341,2181,3091,4702,4732,2173,4513,3524,3165,
2155,1166,3537,3058,3509,2219,1,1321,5,2208,771,1021,3752,
3022,4822,692,623,1233,603,4803,678,4469,4261,4721,2384,1424,
5424,4944,1565,495,1535,4525,1036,4196,1567,47,458,3649,1,5,
3488,1021,3022,692,1233,603,988,4469,4261,5434,2404,1735,1725,
4645,746,3907,4958,3649,4091,1678,1291,2181,4635,4388,521,4931,
3742,3802,443,3723,4328,3012,4243,4913,4429,541,481,164,5015,
5025,3925,3906,3897,4958,3719,3032,5522,513,919,1361,2418,3091,
2181,1291,2155,2145,2878,521,4921,3742,3802,1273,3723,4328,531,
4931,1772,1782,263,1923,478,2342,4763,4393,4913,889,5501,5491,
5611,21,5644,5515,5475,5625,5605,1386,5657,5598,4029,19,2602,
152,5462,4203,1269,4101,1678,1311,3041,3655,3438,1101,4491,
1792,1802,713,1893,878,2342,4763,4393,4429,5501,5481,174,5585,
105,145,136,127,118,1489,2592,5452,4053,669,1321,2418,1291,
2181,3165,4388,521,4921,3742,3802,1273,3723,4328,3012,4243,
4913,4429,1571,1551,5434,1655,1045,3915,5016,5007,3948,3719,
3032,5522,513,4659,5634,2764,2374,305,325,5326,706,277,5247,
288,1698,961,2438,1841,1601,908,4172,4043,4172,1283,4043,3999,
4331,4571,1,4345,4155,3488,4112,1193,5122,2322,1703,1179,4239,
4509,1404,315,5076,5337,288,8,951,4498,4172,1283,4539,1,5,
2998,4112,3283,1179,4509,1698,1951,3601,4438,3621,2788,4172,
3283,4172,1283,1213,3999,3591,3361,2551,4355,4165,3488,4112,
3283,5122,2322,1703,1179,4239,4509,1698,1951,3601,4438,4301,
2338,4172,4403,4043,4172,1713,789,1941,3601,2481,955,4145,3488,
4112,2322,1703,4112,3283,4759,3429,639,8,3611,2968,4172,3283,
4539,3601,5,2998,4112,3283,1179,4509,1351,4488,612,5033,99,
1301,4488,432,5033,4879,1351,4488,612,5033,99,1321,4488,1321,
4488,612,5033,99,612,723,99,1331,4488,4992,5033,4879,1351,
4488,4992,723,4879,1321,4488,612,723,99,2792,333,5254,2366,
417,5118,5299,5579,1,4595,2882,3823,5384,1516,3267,3878,5409,
1599,1,2715,3192,423,1964,2196,297,5228,5299,189,1,4595,1,
4635,3192,423,1964,2196,297,5228,5299,189,1,4615,1,4595,3072,
3963,5394,1526,3837,3298,5409,2089,1,2715,1,2905,1,3685,1,
2935,1,3695,1,3675,1,3675,1,2715,1,2905,1,3695,4475,2318,
3009,2664,3155,3359,2664,4345,2249,5564,3404,1505,1645,3576,
2356,4717,1087,468,3458,1071,4271,2628,1132,4083,3512,1003,
1549,4699,5444,3905,3585,1666,747,468,928,4311,2648,1132,1003,
5109,3989,5554,3854,1665,4785,5006,2636,3907,1097,468,3458,
3458,1061,4311,2628,1051,4291,2628,1132,4033,1132,4033,3512,
2943,3512,3313,1549,1549,4699,4699,5554,3854,4975,4785,5006,
2636,3907,1097,468,3458,3458,1061,4291,2628,1051,4291,2628,
1132,933,1132,4033,3512,3313,3512,3313,1549,1549,4699,4699,
5414,3575,4795,4986,737,468,928,928,4281,2648,1071,2648,1132,
3313,1132,3313,5109,5109,3989,3989,3458,1071,4271,2628,1132,
4083,3512,1003,1939,3939,3458,1611,3371,2828,1132,4083,3512,
1003,1419,3979,4458,4561,961,1688,2032,2223,2032,2073,1439,
3979,4458,941,4541,1688,4062,2233,4062,4373,1479,5099,928,3371,
2858,1132,1003,1419,3979,3458,1061,4311,2628,1132,4033,3512,
2943,4969,5099,3458,3458,1631,3361,2828,3371,1621,2858,1132,
4033,1132,4033,3512,2943,3512,3313,589,1419,3979,769,3458,1051,
4291,2628,1132,4033,3512,2943,1939,3939,3498,4551,1688,2032,
2073,1439,3979,928,3381,2858,1132,2833,1419,3979,2392,2583,
2658,4854,1126,5357,2759,5369,1,3081,1835,4665,3332,1753,3448,
5194,646,77,259,4819,2911,3081,835,3795,2052,3413,2678,2052,
3413,2678,5154,3066,5267,5169,59,574,3066,377,5169,59,2911,
2911,3081,4345,4135,3081,3705,3785,2392,2563,2678,5284,796,87,
1499,2069,2911,3081,2505,2545,3302,1743,2298,364,806,387,1499,
819,2911,3081,3125,3135,2392,2583,2658,4224,796,5347,2759,5379,
1,1,3081,1835,4685,3081,1865,4675,2392,2583,2658,3734,596,
5347,2759,5379,1,1,3081,1875,4685,3081,1865,4665,3332,1753,
3448,344,656,197,259,4219,2911,2911,3081,835,3795,3081,845,
3795,2012,2573,3448,5134,1186,197,259,4219,2911,2911,3081,845,
3765,3081,855,3775,2802,1763,2678,5174,756,67,1499,3849,2911,
2911,3081,2465,2535,3081,3145,3175,2911,1,1885,4665,1,1,3145,
1,1,2895,1,1,1825,4155,1,1,1,3115,1,3105,2911,1,1835,4125,
1,1,1,2515,1,2465,1,1,3665,2911,1,3695,5235,1,1,1,2705,1,
2925,1,1,865,1,1,975,3485,2868,568,2619,1115,4258,4779,2694,
2455,2284,2455,5544,2524,2305,1445,1459,2494,1855,4744,2725,
225,2169,3885,3398,2268,2049,4975,4785,3398,4448,2268,2009,
2049,3865,4368,2268,3478,2258,2009,2009,2049,3565,3398,2268,
2268,2049,1589,505,3398,2268,4448,2268,2009,2009,1589,2848,
2109,2268,3959,2268,3329,3478,3478,2109,559,2268,3959,2268,
2009,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,204,34,995,4625,
4014,4905,4585,214,4004,995,3884,4004,995,4625,4625,1394,5085,
4605,5154,1209,5,3814,5539,5,4864,3279,1209,4585,4585,2094,
5279,1209,4585,4585,5144,1209,684,1209,5,5,4585,354,2139,5539,
5,4585,394,5539,5,5,1904,1209,5,4585,5,1914,5209,1209,4585,
4585,5,5,5184,1209,5,5,5,404,5539,5,5,5,5,5,5,5,5,5,5,
5,5,5,5,5,5,5,5,5046,4837,4888,8,2,3559,3,1259,5046,4837,
4898,8,2,1159,3,1249,5066,4847,4888,8,2,3559,3,1259,5066,
4847,4888,8,2,3559,3,1259,5056,5067,4898,8,2,1159,3,1249,8,
2,3559,3,1259,8,2,3349,3,1259,8,2,3559,3,1259,8,2,3559,3,
1259,8,2,1159,3,1249,8,2,1159,3,1249,8,2,3559,3,1259,8,2,
3559,3,1259,8,2,3349,3,1259,8,2,3349,3,1259,8,2,1159,3,
1249,4348,2989,2478,2989,826,1227,3546,2447,826,3637,826,1987,
3546,1817,2735,5219,1145,5219,4,2424,4585,4,4585,4,4,4,2685,
5,4,4,2685,5,4,5,4,5,5,4,4,5,2685,4,4,5,2685,4,5,5,
4585,2685,2685,4605,4585,4585,2685,4585,4585,5,5,5,5,5,5,
2816,2816,2777,2777,2956,7,6,7,6,7,6,7,6,7,6,7,3468,9,
3478,2749,3478,2749,2278,9,4418,9,1469,3478,2749,3468,9,3478,
2749,3468,8,2278,5319,5319,8,9,4418,1999,249,3468,9,3468,
3468,9,9,2278,8,1999,9,2278,5319,8,9,9,3468,9,8,1999,9,8,
9,9,8,9,9,8,1999,9,2749,5309,9,5309,9,9,2749,2749,9,9,
2749,2749,9,9,9,9,9];
CR_4.codedDecompositionQClass :=
[665,675,555,95,675,345,675,345,95,675,675,96,676,96,676,676,
425,355,425,435,435,365,435,375,465,385,455,475,395,445,465,
405,475,455,475,485,505,525,26,426,36,436,436,26,426,436,36,
426,36,436,436,436,436,66,46,56,46
