\def\x{\times}
\hyphenation{Cryst-Cat-Record}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{The Crystallographic Groups Catalog}

\index{crystallographic groups}
\index{crystcat}

The package {\CrystCat} provides a catalog of crystallographic groups
of dimensions 2, 3, and 4 which covers most of the data contained 
in the book ``Crystallographic groups of four-dimensional space''
\cite{BBNWZ78}. It has been brought into {\GAP} format by Volkmar
Felsch.

The {\GAP} 4 version of the catalog requires the package {\Cryst},
which is loaded automatically. The benefit of this is that space groups 
extracted from the catalog now have the rich set of methods provided
by {\Cryst} at their disposal, and are no longer dumb lists of generators.
Moreover, space groups are now fully supported in both the representation
acting from the left and the representation acting from the right.

In 2001, Bernd Souvignier has discovered an error in the above mentioned
book: On page 118, in the tabulation of enantiomorphic space-group types,
it is wrongly claimed that the (affine) four-dimensional space-group
type 08/01/01/002 splits into an enantiomorphic pair of (proper)
space-group types. This is indicated by an asterisk preceding the
space-group number. This asterisk has to be removed. As a
consequence, the number of four-dimensional space-group types
splitting into enantiomorphic pairs (given on page 11 and page 52
of the book) reduces from 112 to 111. An erratum has been submitted
to {\it Acta Cryst.}.

The only implication of this correction for the package {\CrystCat}
is that the output of the function
\begintt
DisplaySpaceGroupType( 4, 8, 1, 1, 2 );
\endtt
had to be changed from
\begintt
#I    *Space-group type (4,8,1,1,2); orbit size 2; fp-free
\endtt
to
\begintt
#I     Space-group type (4,8,1,1,2); orbit size 2; fp-free
\endtt
This has been done in the release {\GAP} 4.3.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{How to access the data of the book}

Among others, the catalog offers functions which provide access to the
data listed in the Tables 1, 5, and 6 of \cite{BBNWZ78}:

\beginlist%unordered 
\item{$\bullet$} 
    The information on the crystal families listed in Table 1 can
    be reproduced using the `DisplayCrystalFamily' function.
\item{$\bullet$} 
    Similarly, the `DisplayCrystalSystem' function can be
    used to reproduce the information on the crystal systems
    provided in Table 1.
\item{$\bullet$} 
    The information given in the $\Q$-class headlines of
    Table 1 can be displayed by the `DisplayQClass' function,
    whereas the `FpGroupQClass' function can be used to reproduce
    the presentations that are listed in Table 1 for the $\Q$-class
    representatives.
\item{$\bullet$} 
    The information given in the $\Z$-class headlines of Table 1
    will be covered by the results of the `DisplayZClass' function,
    and the matrix generators of the $\Z$-class representatives can
    be constructed by calling the `MatGroupZClass' function.
\item{$\bullet$} 
    The `DisplaySpaceGroupType' and the
    `DisplaySpaceGroupGenerators' functions can be used to
    reproduce all of the information on the space-group types that
    is provided in Table 1.
\item{$\bullet$} 
    The normalizers listed in Table 5 can be reproduced by
    calling the `NormalizerZClass' function.
\item{$\bullet$} 
    Finally, the `CharTableQClass' function will compute the
    character tables listed in Table 6, whereas the isomorphism
    types given in Table 6 may be obtained by calling the
    `DisplayQClass' function.  
\endlist

The display functions mentioned in the above list print their output
with different indentation.  So, calling them in a suitably nested
loop, you may produce a listing in which the information about the
objects of different type will be properly indented as has been done
in Table 1 of \cite{BBNWZ78}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Representation of space groups}

Probably the most important function in the catalog is the
`SpaceGroupBBNWZ' function which provides representatives of the
affine classes of space groups.  A space group of dimension $n$ is
represented by an $(n+1)$-dimensional rational matrix group as
follows.

If $S$ is an $n$-dimensional space group, then each element $s$
in $S$ is an affine mapping $s: V \to V$ of an $n$-dimensional 
$\R$-vector space $V$ onto itself.  Hence $s$ can be written as 
the product of an appropriate invertible linear mapping
$g: V \to V$ and a translation by some translation
vector $t \in V$ such that, if we write mappings from the left, 
we have $s(v) = g(v) + t$ for all $v \in V$.

If we fix a basis of $V$ and then replace each $v \in V$ by the column
vector of its coefficients with respect to that basis (and hence $V$
by the isomorphic column vector space $\R^{n \x 1}$), we can describe
the linear mapping $g$ involved in $s$ by an $n \x n$ matrix
$M_g \in GL_n(\R)$ which acts by multiplication from the left on
the column vectors in $\R^{n \x 1}$.  Hence, if we identify $V$ with
$\R^{n \x 1}$, we have $s(v) = M_g v + t$ for all 
$v \in \R^{n \x 1}$.

Moreover, if we extend each column vector $v \in \R^{n \x 1}$ to a
column $[ [ v ], [ 1 ] ]$ of length $n+1$ by adding an entry 1 
in the last position and if we define an $(n+1) \x (n+1)$ matrix 
$M_s = [ [ M_g, t ], [ 0, 1 ] ]$, we have $[ [ s(v) ], [ 1 ] ] = 
M_s [ [ v ], [ 1 ] ]$ for all $v \in \R^{n \x 1}$. This means that 
we can represent the space group $S$ by the isomorphic group 
$M(S) = \{ M_s | s \in S \}$.  The submatrices $M_g$ occurring in 
the elements of $M(S)$ form an $n \x n$ matrix group $P(S)$, the 
``point group'' of $M(S)$.  In fact, we can choose the basis of 
$\R^{n \x 1}$ such that $M_g \in GL_n(\Z)$ and  $t \in \Q^{n \x 1}$ 
for all $M_s \in M(S)$.  In particular, the space group 
representatives that are normally used by the crystallographers
are of this form, and the book \cite{BBNWZ78} uses the same convention.

The representation described above is the one usually used by
crystallographers. There is, however, an alternative to the
representation of the space group elements by matrices of the form
$[ [ M_g, t ], [ 0, 1 ] ]$ as described above. Instead of 
considering the coefficient vectors as columns we may consider 
them as rows.  Then we can associate to each affine mapping
$s \in S$ an $(n+1) \x (n+1)$ matrix $M\pif_s =
[ [ M\pif_{g\pif}, 0 ], [ t\pif, 1 ] ]$ with 
$M\pif_{g\pif} \in GL_n(\R)$ and $t\pif \in \R^{1 \x n}$ such 
that $[s(v\pif),1] = [v\pif,1] M\pif_s$ for all 
$v\pif \in \R^{1 \x n}$, and we may represent $S$ by the matrix 
group $M\pif(S) = \{ M\pif_s | s \in S \}$.  Again, we can choose 
the basis of $\R^{1 \x n}$ such that $M\pif_{g\pif} \in GL_n(\Z)$ 
and $t\pif \in \Q^{1 \x n}$ for all $M\pif_s \in M\pif(S)$.

From the mathematical point of view, both approaches are equivalent. 
In particular, $M(S)$ and $M\pif(S)$ are isomorphic, for instance 
via the isomorphism $\tau$ mapping $M_s \in M(S)$ to 
$(M_s^{tr})^{-1}$.  Unfortunately, however, neither of the two 
is a good choice for our {\GAP} catalog.

The first convention, using matrices which act on column vectors from
the left, is not consistent with the fact that actions in {\GAP} are
usually from the right.

On the other hand, if we choose the second convention, we run into a
problem with the names of the space groups as introduced in
\cite{BBNWZ78}. Any such name does not just describe the abstract
isomorphism type of the respective space group $S$, but reflects
properties of the matrix group $M(S)$.  In particular, it contains as
a leading part the name of the $\Z$-class of the associated point
group $P(S)$.  Since the classification of space groups by affine
equivalence is tantamount to their classification by abstract
isomorphism, $M\pif(S)$ lies in the same affine class as $M(S)$ and
hence should get the same name as $M(S)$.  But the point group $P(S)$
that occurs in that name is not always $\Z$-equivalent to the point
group $P\pif(S)$ of $M\pif(S)$.  For example, the isomorphism $\tau:
M(S) \to M\pif(S)$ defined above maps the $\Z$-class representative
with the parameters $[3,7,3,2]$ (in the notation described below) to
the $\Z$-class representative with the parameters $[3,7,3,3]$.  In
other words:\ The space group names introduced for the groups $M(S)$
in \cite{BBNWZ78} lead to confusing inconsistencies if assigned to the
groups $M\pif(S)$.

In order to avoid this confusion we decided that the first convention
is the lesser evil, and so the {\GAP} catalog follows the book. In
particular, all functions listed in section "How to access the data 
of the book" use the convention of the book. The space groups, however,
can be constructed in both representations, so that the user can choose
the one that seems more appropriate in the particular situation.
The function `SpaceGroupOnLeftBBNWZ' constructs a space group in
the ``crystallographic'' representation acting on the left,
whereas `SpaceGroupOnRightBBNWZ' constructs a space group in the
representation acting on the right, as preferred by GAP. In order to 
avoid long function names (and in order to avoid mixing groups in
different representations), one can set one{\pif}s own default 
with the function `SetCrystGroupDefaultAction' 
(see "ref:SetCrystGroupDefaultAction"), 
which takes as argument either `LeftAction' of `RightAction'. 
`SpaceGroupBBNWZ' then constructs a space group in this default
representation. Initially, the default is `RightAction'.

The space groups constructed from the catalog are matrix groups, which
in addition have the property `IsAffineCrystGroupOnLeft' (or
`IsAffineCrystGroupOnRight', respectively). The package {\Cryst}
provides methods to compute with such groups. {\Cryst} is necessary
for any serious computation with space groups, because the support of
plain {\GAP} for infinite matrix groups (such as space groups) is very
limited.

Before we describe all available catalog functions in detail, we have
to add two remarks.

*Remark 1:*
The concepts  used  in  this section are   defined  in chapter 1   (Basic
definitions) of \cite{BBNWZ78}.  However, note that the definition of the
concept of a crystal system given on page  16 of that  book relies on the
following statement about $\Q$-classes:

\beginitems
& For a  $\Q$-class  <C>  there is a unique holohedry <H> such that
  each  f.u. group in <C>  is a subgroup of some  f.u. group in
  <H>,  but is not  a subgroup of any  f.u. group  belonging to a
  holohedry of smaller order.
\enditems

This  statement is correct for  dimensions 1, 2,  3, and 4, and hence the
definition of ``crystal system'' given on page 16 of \cite{BBNWZ78} is
known  to   be unambiguous for   these  dimensions.  However, there  is a
counterexample to  this statement in seven-dimensional  space so that the
definition breaks down for some higher dimensions.

Therefore,  the authors of  the book have  since proposed to replace this
definition of ``crystal system''  by the  following much simpler  one,
which has been discussed in more detail in \cite{NPW81}. To formulate it,
we use    the   intersections  of $\Q$-classes    and  Bravais  flocks as
introduced on page 17 of \cite{BBNWZ78}, and we define the classification
of the set of all $\Z$-classes into crystal systems as follows:

\beginitems
& *Definition*:\  A crystal system  (introduced as an  equivalence
  class  of   $\Z$-classes)   consists of  full  geometric  crystal
  classes.  The $\Z$-classes  of  two  (geometric)  crystal classes
  belong to the same crystal system  if and only if these geometric
  crystal  classes  intersect  the same  set of  Bravais  flocks of
  $\Z$-classes.
\enditems

From  this  definition  of  a  crystal system  of  $\Z$-classes  one then
obtains  crystal systems of f.u.  groups,  of space-group types, and of
space groups in the same manner as with the  preceding definitions in the
book.

The new definition is unambiguous for all dimensions. Moreover, it can be
checked  from  the   tables   in the  book   that  it   defines  the same
classification as the old one for dimensions 1, 2, 3, and 4.

It should  be noted that  the concept of  crystal  family is well-defined
independently of the dimension if one uses the ``more natural'' second
definition of it at the end of page 17. Moreover, the first definition of
crystal family on page  17 defines the same concept  as the second one if
the now proposed definition of crystal system is used.


*Remark 2:*
The second remark just concerns a  different terminology in the tables of
\cite{BBNWZ78} and in the current catalog. In group theory, the number of
elements of a  finite group  usually is  called  the ``order'' of  the
group.   This notation has  been  used throughout   in  the book.   Here,
however,  we  will follow the    {\GAP}   conventions and use  the   term
``size'' instead.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Crystal Families}

\>NrCrystalFamilies( <dim> )

returns the number of crystal families in case of dimension <dim>.  It
can be used to formulate loops over the crystal families.

There are 4, 6, and 23 crystal families of dimension 2, 3, and 4,
respectively.

\beginexample
gap> n := NrCrystalFamilies( 4 );
23
\endexample

\>DisplayCrystalFamily( <dim>, <family> )

displays for the specified crystal family essentially the same
information as is provided for that family in Table 1 of
\cite{BBNWZ78}, namely

\beginlist%unordered
\item{$\bullet$} the family name,

\item{$\bullet$} the number of parameters,

\item{$\bullet$} the common rational decomposition pattern,

\item{$\bullet$} the common real decomposition pattern,

\item{$\bullet$} the number of crystal systems in the family, and

\item{$\bullet$} the number of Bravais flocks in the family.
\endlist

For details see \cite{BBNWZ78}.

\beginexample
gap> DisplayCrystalFamily( 4, 17 );
#I Family XVII: cubic orthogonal; 2 free parameters;
#I  Q-decomposition pattern 1+3; R-decomposition pattern 1+3;
#I  2 crystal systems; 6 Bravais flocks
gap> DisplayCrystalFamily( 4, 18 );
#I Family XVIII: octagonal; 2 free parameters;
#I  Q-irreducible; R-decomposition pattern 2+2;
#I  1 crystal system; 1 Bravais flock
gap> DisplayCrystalFamily( 4, 21 );
#I Family XXI: di-isohexagonal orthogonal; 1 free parameter;
#I  R-irreducible; 2 crystal systems; 2 Bravais flocks
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Crystal Systems}

\>NrCrystalSystems( <dim> )

returns the number of crystal systems in case of dimension <dim>. It
can be used to formulate loops over the crystal systems.

There are 4, 7, and 33 crystal systems of dimension 2, 3, and 4,
respectively.

\beginexample
gap> n := NrCrystalSystems( 2 );
4
\endexample

The following two functions are functions of crystal systems.

Each crystal system is characterized by a pair (<dim>, <system>) where
<dim>  is the associated dimension,  and  <system> is  the number of  the
crystal system.

\>DisplayCrystalSystem( <dim>, <system> )

displays for the specified crystal system essentially the same
information as is provided for that system in Table 1 of
\cite{BBNWZ78}, namely

\beginlist%unordered
\item{$\bullet$} 
    the number of $\Q$-classes in the crystal system and
\item{$\bullet$} 
    the identification number, i.~e., the triple (<dim>, <system>,
    <q-class>) described below, of the $\Q$-class that is the 
    holohedry of the crystal system.  
\endlist

For details see \cite{BBNWZ78}.

\beginexample
gap> for sys in [ 1 .. 4 ] do  DisplayCrystalSystem( 2, sys );  od;
#I  Crystal system 1: 2 Q-classes; holohedry (2,1,2)
#I  Crystal system 2: 2 Q-classes; holohedry (2,2,2)
#I  Crystal system 3: 2 Q-classes; holohedry (2,3,2)
#I  Crystal system 4: 4 Q-classes; holohedry (2,4,4)
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Q-Classes}

\>NrQClassesCrystalSystem( <dim>, <system> )

returns the number of $\Q$-classes within the given crystal system.
It can be used to formulate loops over the $\Q$-classes.

The following five functions are functions of $\Q$-classes.

In general, the parameters characterizing a $\Q$-class will form a
triple (<dim>, <system>, <q-class>) where <dim> is the associated
dimension, <system> is the number of the associated crystal system,
and <q-class> is the number of the $\Q$-class within the crystal
system.  However, in case of dimensions 2 or 3, a $\Q$-class may also
be characterized by a pair (<dim>, <IT-number>) where <IT-number> is
the number in the International Tables for Crystallography
\cite{Hah95} of any space-group type lying in (a $\Z$-class of) that
$\Q$-class, or just by the Hermann-Mauguin symbol of any space-group
type lying in (a $\Z$-class of) that $\Q$-class.

The Hermann-Mauguin symbols \index{Hermann-Mauguin symbol} which we
use in {\GAP} are the short Hermann-Mauguin symbols defined in the
1983 edition of the International Tables \cite{Hah95}, but any
occurring indices are expressed by ordinary integers, and bars are
replaced by minus signs.  For example, the Hermann-Mauguin symbol
<P>$\overline{4}2_1m$ will be represented by the string `\"P-421m\"'.

\>DisplayQClass( <dim>, <system>, <q-class> )
\>DisplayQClass( <dim>, <IT-number> )
\>DisplayQClass( <Hermann-Mauguin-symbol> )

displays for the specified $\Q$-class essentially the same information
as is provided for that $\Q$-class in Table 1 of \cite{BBNWZ78}
(except for the defining relations given there), namely
\beginlist%unordered
\item{$\bullet$} the size of the groups in the $\Q$-class,
\item{$\bullet$} the isomorphism type of the groups in the $\Q$-class,
\item{$\bullet$} the Hurley pattern,
\item{$\bullet$} the rational constituents,
\item{$\bullet$} the number of $\Z$-classes in the $\Q$-class, and
\item{$\bullet$} the number of space-group types in the $\Q$-class.
\endlist
For details see \cite{BBNWZ78}.

\beginexample
gap> DisplayQClass( "p2" );
#I   Q-class H (2,1,2): size 2; isomorphism type 2.1 = C2;
#I    Q-constituents 2*(2,1,2); cc; 1 Z-class; 1 space group
gap> DisplayQClass( "R-3" );
#I   Q-class (3,5,2): size 6; isomorphism type 6.1 = C6;
#I    Q-constituents (3,1,2)+(3,4,3); ncc; 2 Z-classes; 2 space grps
gap> DisplayQClass( 3, 195 );
#I   Q-class (3,7,1): size 12; isomorphism type 12.5 = A4;
#I    C-irreducible; 3 Z-classes; 5 space grps
gap> DisplayQClass( 4, 27, 4 );
#I   Q-class H (4,27,4): size 20; isomorphism type 20.3 = D10xC2;
#I    Q-irreducible; 1 Z-class; 1 space group
gap> DisplayQClass( 4, 29, 1 );
#I  *Q-class (4,29,1): size 18; isomorphism type 18.3 = D6xC3;
#I    R-irreducible; 3 Z-classes; 5 space grps
\endexample

Note in the preceding examples that, as pointed out above, the term
``size'' denotes the order of a representative group of the specified
$\Q$-class and, of course, not the (infinite) class length.

\>FpGroupQClass( <dim>, <system>, <q-class> )
\>FpGroupQClass( <dim>, <IT-number> )
\>FpGroupQClass( <Hermann-Mauguin-symbol> )

returns a finitely presented group <F>, say, which is isomorphic to
the groups in the specified $\Q$-class.

The presentation of that group is the same as the corresponding
presentation given in Table 1 of \cite{BBNWZ78} except for the fact
that its generators are listed in reverse order.  The reason for this
change is that, whenever the group in question is solvable, the
resulting generators form a pcgs (as defined in section "ref:Polycyclic
Groups" in the reference manual of {\GAP}) if they are numbered 
``from the top to the bottom'', and the presentation is a 
power-commutator presentation.  The `PcGroupQClass' function described 
next will make use of this fact in order to construct a pc group 
isomorphic to <F>.

Note that, for any $\Z$-class  in the  specified $\Q$-class,  the  matrix
group  returned by the `MatGroupZClass'  function (see below) not only is
isomorphic to <F>, but also its  generators satisfy the defining relators
of <F>.

Besides  of the usual components,  <F> will have an attribute
`CrystCatRecord', which is a record with component
`parameters', which keeps a list of the parameters
that specify the given $\Q$-class.

\beginexample
gap> F := FpGroupQClass( 4, 20, 3 );
FpGroupQClass( 4, 20, 3 )
gap> GeneratorsOfGroup( F );
[ f1, f2 ]
gap> RelatorsOfFpGroup( F );
[ f1^2*f2^-3, f2^6, f2^-1*f1^-1*f2*f1*f2^-4 ]
gap> Size( F );
12
gap> CrystCatRecord( F ).parameters;
[ 4, 20, 3 ]
\endexample

\>PcGroupQClass( <dim>, <system>, <q-class> )
\>PcGroupQClass( <dim>, <IT-number> )
\>PcGroupQClass( <Hermann-Mauguin-symbol> )

returns a pc group $P$, say, isomorphic to the groups in the
specified $\Q$-class, if these groups are solvable, or the value
`fail' (together with an appropriate warning), otherwise.

$P$ is constructed by first establishing a finitely presented group
(as it would be returned by the `FpGroupQClass' function described
above) and then constructing from it an isomorphic pc group.  If the
underlying pcgs is not a prime orders pcgs (see section "ref:Polycyclic
Groups"), then it will be refined appropriately (and a warning will 
be displayed).

Besides of the usual components, <P> will have an attribute
`CrystCatRecord', which is a record with component
`parameters', which saves a list of the parameters that specify 
the given $\Q$-class.

\beginexample
gap> P := PcGroupQClass( 4, 31, 3 );
#I  Warning: a non-solvable group can't be represented as a pc group
fail
gap> P := PcGroupQClass( 4, 20, 3 );
PcGroupQClass( 4, 20, 3 )
gap> GeneratorsOfGroup( P );
[ f1, f2 ]
gap> Size( P );
12
gap> CrystCatRecord( P ).parameters;
[ 4, 20, 3 ]
\endexample

\>CharTableQClass( <dim>, <system>, <q-class> )
\>CharTableQClass( <dim>, <IT-number> )
\>CharTableQClass( <Hermann-Mauguin-symbol> )

returns the character table $T$, say, of a representative group of (a
$\Z$-class of) the specified $\Q$-class.

Although the set of characters  can be considered as  an invariant of the
specified $\Q$-class,  the resulting  table  will depend on  the order in
which {\GAP} sorts the conjugacy classes of  elements and the irreducible
characters  and    hence,  in  general,   will  not   coincide  with  the
corresponding table presented in \cite{BBNWZ78}.

`CharTableQClass' proceeds as follows.  If the groups in the given
$\Q$-class are solvable, then it first calls the `PcGroupQClass' and
`RefinedPcGroup' functions to get a suitable isomorphic pc group, and 
then it calls the `CharacterTable' function to compute the
character table of that pc group.  In the case of the five
$\Q$-classes of dimension 4 whose groups are not solvable, it first
calls the `FpGroupQClass' function to get an isomorphic finitely
presented group, then it constructs a specially chosen faithful
permutation representation of low degree for that group, and finally
it determines the character table of the resulting permutation group
again by calling the `CharacterTable' function.

In general, the above strategy will be much more efficient than the
alternative possibilities of calling the `CharacterTable' function for a
finitely presented group provided by the `FpGroupQClass' function or
for a matrix group provided by the `MatGroupZClass' function.

\beginexample
gap> T := CharTableQClass( 4, 20, 3 );;
gap> Display( T );
CharTableQClass( 4, 20, 3 )

     2  2  1  1  2  2  2
     3  1  1  1  1  .  .

       1a 3a 6a 2a 4a 4b
    1P 1a 3a 2a 6a 4a 4b
    2P 1a 3a 3a 1a 2a 2a
    3P 1a 1a 2a 2a 4b 4a
    5P 1a 3a 6a 2a 4a 4b

X.1     1  1  1  1  1  1
X.2     1  1  1  1 -1 -1
X.3     1  1 -1 -1  A -A
X.4     1  1 -1 -1 -A  A
X.5     2 -1  1 -2  .  .
X.6     2 -1 -1  2  .  .

A = E(4)
  = ER(-1) = i
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Z-Classes}

\>NrZClassesQClass( <dim>, <system>, <q-class> )
\>NrZClassesQClass( <dim>, <IT-number> )
\>NrZClassesQClass( <Hermann-Mauguin-symbol> )

returns the number of $\Z$-classes within the given $\Q$-class. It can
be used to formulate loops over the $\Z$-classes.

The following functions are functions of $\Z$-classes.

In general, the parameters characterizing a $\Z$-class will form a
quadruple (<dim>, <system>, <q-class>, <z-class>) where <dim> is the
associated dimension, <system> is the number of the associated crystal
system, <q-class> is the number of the associated $\Q$-class within
the crystal system, and <z-class> is the number of the $\Z$-class
within the $\Q$-class.  However, in case of dimensions 2 or 3, a
$\Z$-class may also be characterized by a pair (<dim>, <IT-number>)
where <IT-number> is the number in the International Tables
\cite{Hah95} of any space-group type lying in that $\Z$-class, or just
by the Hermann-Mauguin symbol of any space-group type lying in that
$\Z$-class.

\>DisplayZClass( <dim>, <system>, <q-class>, <z-class> )
\>DisplayZClass( <dim>, <IT-number> )
\>DisplayZClass( <Hermann-Mauguin-symbol> )

displays for the specified $\Z$-class essentially the same information
as is provided for that $\Z$-class in Table 1 of \cite{BBNWZ78}
(except for the generating matrices of a class representative group
given there), namely
\beginlist%unordered
\item{$\bullet$} for  dimensions 2 and 3, the  Hermann-Mauguin symbol of a
    representative space-group type  which belongs  to that $\Z$-class,
\item{$\bullet$} the Bravais type,
\item{$\bullet$} some decomposability information,
\item{$\bullet$} the number of space-group types belonging to the $\Z$-class,
\item{$\bullet$} the size of the associated cohomology group.
\endlist
For details see \cite{BBNWZ78}.

\beginexample
gap> DisplayZClass( 2, 3 );
#I    Z-class (2,2,1,1) = Z(pm): Bravais type II/I; fully Z-reducible;
#I     2 space groups; cohomology group size 2
gap> DisplayZClass( "F-43m" );
#I    Z-class (3,7,4,2) = Z(F-43m): Bravais type VI/II; Z-irreducible;
#I     2 space groups; cohomology group size 2
gap> DisplayZClass( 4, 2, 3, 2 );
#I    Z-class B (4,2,3,2): Bravais type II/II; Z-decomposable;
#I     2 space groups; cohomology group size 4
gap> DisplayZClass( 4, 21, 3, 1 );
#I   *Z-class (4,21,3,1): Bravais type XVI/I; Z-reducible;
#I     1 space group; cohomology group size 1
\endexample

\>MatGroupZClass( <dim>, <system>, <q-class>, <z-class> )
\>MatGroupZClass( <dim>, <IT-number> )
\>MatGroupZClass( <Hermann-Mauguin-symbol> )

returns a $dim \x dim$ matrix group <M>, say, which is a
representative of the specified $\Z$-class.  Its generators satisfy
the defining relators of the finitely presented group which may be
computed by calling the `FpGroupQClass' function (see above) for the
$\Q$-class which contains the given $\Z$-class.

The generators of <M> are the same matrices as those given  in Table 1 of
\cite{BBNWZ78}. Note, however, that they will  be listed in reverse order
to  keep them  in parallel  to  the abstract  generators provided by  the
`FpGroupQClass' function (see above).

Besides of the usual components, <M> will have an attribute
`CrystCatRecord', which is a record with two components.  The
first component is `parameters', which saves a list of the parameters
that specify the given $\Z$-class. The second component is
`conjugator', whose value is the identity element of <M>. Its purpose
is to make the resulting record consistent with those returned by
the `NormalizerZClass' or `ZClassRepsDadeGroup' functions described
below.

\beginexample
gap> M := MatGroupZClass( 4, 20, 3, 1 );
MatGroupZClass( 4, 20, 3, 1 )
gap> for g in GeneratorsOfGroup( M ) do
>  Print( "\n" ); PrintArray( g ); od; Print( "\n" );

[ [   0,   1,   0,   0 ],
  [  -1,   0,   0,   0 ],
  [   0,   0,  -1,  -1 ],
  [   0,   0,   0,   1 ] ]

[ [  -1,   0,   0,   0 ],
  [   0,  -1,   0,   0 ],
  [   0,   0,  -1,  -1 ],
  [   0,   0,   1,   0 ] ]

gap> Size( M );
12
gap> CrystCatRecord( M ).parameters;
[ 4, 20, 3, 1 ]
\endexample

\>NormalizerZClass( <dim>, <system>, <q-class>, <z-class> )
\>NormalizerZClass( <dim>, <IT-number> )
\>NormalizerZClass( <Hermann-Mauguin-symbol> )

returns the normalizer <N>, say, in $GL(dim,\Z)$ of the representative
$dim \x dim$ matrix group which is constructed by the
`MatGroupZClass' function (see above).

If the size of <N> is finite, then <N> again lies in some $\Z$-class.
In this case, <N> will have an attribute `CrystCatRecord',
which is a record with two components, `parameters' and `conjugator'.
These contain, respectively, the list of parameters of that
$\Z$-class, and a matrix $g \in GL(dim,\Z)$, such that $N = g^{-1} R g$,
where $R$ is the representative group of that $\Z$-class.

\beginexample
gap> N := NormalizerZClass( 4, 20, 3, 1 );
NormalizerZClass( 4, 20, 3, 1 )
gap> for g in GeneratorsOfGroup( N ) do
>  Print( "\n" ); PrintArray( g ); od; Print( "\n" );

[ [   1,   0,   0,   0 ],
  [   0,   1,   0,   0 ],
  [   0,   0,   1,   0 ],
  [   0,   0,  -1,  -1 ] ]

[ [   1,   0,   0,   0 ],
  [   0,  -1,   0,   0 ],
  [   0,   0,  -1,  -1 ],
  [   0,   0,   1,   0 ] ]

[ [   0,   1,   0,   0 ],
  [  -1,   0,   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 ] ]

gap> Size( N );
96
gap> CrystCatRecord( N ).parameters;
[ 4, 20, 22, 1 ]
gap> CrystCatRecord( N ).conjugator = One( N );
true
gap> L := NormalizerZClass( 3, 42 );
NormalizerZClass( 3, 3, 2, 4 )
gap> Size( L );
16
gap> CrystCatRecord( L ).parameters;
[ 3, 4, 7, 2 ]
gap> CrystCatRecord( L ).conjugator;
[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, -1 ] ]
gap> M := NormalizerZClass( "C2/m" );
<matrix group of size infinity with 5 generators>
gap> Size( M );
"infinity"
gap> HasCrystCatRecord( M );
false
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Dade groups}

Some of the $\Z$-classes  of   dimension <d>, say, are ``maximal''  in
the sense  that the groups in these  classes are maximal finite subgroups
of $GL(d,\Z)$.  Generalizing a term which is  being used for dimension 4,
we call the representatives   of these  maximal $\Z$-classes  the *Dade
groups* of dimension $d$.

\>NrDadeGroups( <dim> )

returns the number of Dade groups of dimension <dim>. It can be used
to formulate loops over the Dade groups.

There are 2, 4, and 9 Dade groups of dimension 2, 3, and 4, respectively.

\beginexample
gap> NrDadeGroups( 4 );
9
\endexample

\>DadeGroup( <dim>, <n> )

returns the <n>th Dade group of dimension <dim>.

\beginexample
gap> D := DadeGroup( 4, 7 );
MatGroupZClass( 4, 31, 7, 2 )
\endexample

\>DadeGroupNumbersZClass( <dim>, <system>, <q-class>, <z-class> )
\>DadeGroupNumbersZClass( <dim>, <IT-number> )
\>DadeGroupNumbersZClass( <Hermann-Mauguin-symbol> )

returns the set of all those integers $n_i$ for which the $n_i$th Dade
group of dimension <dim> contains a subgroup which, in $GL(dim,\Z)$,
is conjugate to the representative group of the given $\Z$-class.

\beginexample
gap> dadeNums := DadeGroupNumbersZClass( 4, 4, 1, 2 );
[ 1, 5, 8 ]
gap> for d in dadeNums do
>     D := DadeGroup( 4, d );
>     Print( D, " of size ", Size( D ), "\n" );
> od;
MatGroupZClass( 4, 20, 22, 1 ) of size 96
MatGroupZClass( 4, 30, 13, 1 ) of size 288
MatGroupZClass( 4, 32, 21, 1 ) of size 384
\endexample

\>ZClassRepsDadeGroup( <dim>, <system>, <q-class>, <z-class>, <n> )
\>ZClassRepsDadeGroup( <dim>, <IT-number>, <n> )
\>ZClassRepsDadeGroup( <Hermann-Mauguin-symbol>, <n> )

determines in the <n>th Dade group of dimension <dim> all those
conjugacy classes whose groups are, in $GL(dim,\Z)$, conjugate to the
$\Z$-class representative group <R>, say, of the given $\Z$-class.  It
returns a list of representative groups of these conjugacy classes.

Let <M> be any group in the resulting list.  <M> then has an attribute
`CrystCatRecord', which is a record with two components. The
component `parameters' is the list of parameters of the $\Z$-class of
<R>, and `conjugator' is a suitable matrix <g> from $GL(dim,\Z)$,
respectively, such that <M> equals $g^{-1} R g$.

\beginexample
gap> DadeGroupNumbersZClass( 2, 2, 1, 2 );
[ 1, 2 ]
gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 1 );
[ MatGroupZClass( 2, 2, 1, 2 )^[ [ 0, 1 ], [ -1, 0 ] ] ]
gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 2 );
[ MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, -1 ], [ 0, -1 ] ],
  MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, 0 ], [ -1, 1 ] ] ]
gap> R := last[2];;
gap> CrystCatRecord( R ).parameters;
[ 2, 2, 1, 2 ]
gap> CrystCatRecord( R ).conjugator;
[ [ 1, 0 ], [ -1, 1 ] ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Space groups and space group types}

\>NrSpaceGroupTypesZClass( <dim>, <system>, <q-class>, <z-class> )
\>NrSpaceGroupTypesZClass( <dim>, <IT-number> )
\>NrSpaceGroupTypesZClass( <Hermann-Mauguin-symbol> )

returns the number of space-group types within the given $\Z$-class.
It can be used to formulate loops over the space-group types.

\beginexample
gap> N := NrSpaceGroupTypesZClass( 4, 4, 1, 1 );
13
\endexample

The following functions are functions of space-group types.

In general, the parameters characterizing a space-group type will form
a quintuple (<dim>, <system>, <q-class>, <z-class>, <sg-type>) where
<dim> is the associated dimension, <system> is the number of the
associated crystal system, <q-class> is the number of the associated
$\Q$-class within the crystal system, <z-class> is the number of the
$\Z$-class within the $\Q$-class, and <sg-type> is the space-group
type within the $\Z$-class.  However, in case of dimensions 2 or 3,
you may instead specify a $\Z$-class by a pair (<dim>, <IT-number>)
or by its Hermann-Mauguin symbol (as described above).  Then the
function will handle the first space-group type within that
$\Z$-class, i.e., <sg-type> = 1, that is, the corresponding
symmorphic space group (split extension).

\>DisplaySpaceGroupType( <dim>, <system>, <q-class>, <z-class>, <sg-type> )
\>DisplaySpaceGroupType( <dim>, <IT-number> )
\>DisplaySpaceGroupType( <Hermann-Mauguin-symbol> )

displays for the specified space-group type some of the information
which is provided for that space-group type in Table 1 of
\cite{BBNWZ78}, namely
\beginlist%unordered
\item{$\bullet$} the orbit size associated with that space-group type and,
\item{$\bullet$} for dimensions  2 and 3,  the <IT-number>  and  the 
    Hermann-Mauguin symbol.
\endlist
For details see \cite{BBNWZ78}.

\beginexample
gap> DisplaySpaceGroupType( 2, 17 );
#I     Space-group type (2,4,4,1,1); IT(17) = p6mm; orbit size 1
gap> DisplaySpaceGroupType( "Pm-3" );
#I     Space-group type (3,7,2,1,1); IT(200) = Pm-3; orbit size 1
gap> DisplaySpaceGroupType( 4, 32, 10, 2, 4 );
#I    *Space-group type (4,32,10,2,4); orbit size 18
gap> DisplaySpaceGroupType( 3, 6, 1, 1, 4 );
#I    *Space-group type (3,6,1,1,4); IT(169) = P61, IT(170) = P65;
#I      orbit size 2; fp-free
\endexample

\>DisplaySpaceGroupGenerators( <dim>, <system>, <q-class>, <z-class>, <sg-type>)
\>DisplaySpaceGroupGenerators( <dim>, <IT-number> )
\>DisplaySpaceGroupGenerators( <Hermann-Mauguin-symbol> )

displays the non-translation generators of a representative space
group of the specified space-group type without actually constructing
that matrix group. The generators are given in the representation
acting from the left on column vectors.

In more details: Let <n> = <dim> be the given dimension, and let $M_1,
\ldots, M_r$ be the generators of the representative $n \x n$
matrix group of the given $\Z$-class (this is the group which you will
get if you call the `MatGroupZClass' function (see above) for that
$\Z$-class).  Then, for the given space-group type, the 
`SpaceGroupOnLeftBBNWZ' function described below will construct as 
representative of that space-group type an $(n+1) \x (n+1)$ matrix 
group which is generated by the <n> translations which are induced 
by the (standard) basis vectors of the <n>-dimensional Euclidian space, 
and <r> additional matrices $S_1, \ldots, S_r$ of the form $S_i =
\left[\matrix{ M_i & t_i \cr 0 & 1 } \right]$, where the $n \x n$
submatrices $M_i$ are as defined above, and the $t_i$ are <n>-columns
with rational entries.  The `DisplaySpaceGroupGenerators' function
saves time by not constructing the group, but just displaying the $r$
matrices $S_1,\ldots, S_r$.

\beginexample
gap> DisplaySpaceGroupGenerators( "P61" );
#I  Non-translation generators of SpaceGroupOnLeftBBNWZ( 3, 6, 1, 1, 4 )

[ [   -1,    0,    0,    0 ],
  [    0,   -1,    0,    0 ],
  [    0,    0,    1,  1/2 ],
  [    0,    0,    0,    1 ] ]

[ [    0,   -1,    0,    0 ],
  [    1,   -1,    0,    0 ],
  [    0,    0,    1,  1/3 ],
  [    0,    0,    0,    1 ] ]

\endexample

\>SpaceGroupOnLeftBBNWZ( <dim>, <system>, <q-class>, <z-class>, <sg-type> )
\>SpaceGroupOnLeftBBNWZ( <dim>, <IT-number> )
\>SpaceGroupOnLeftBBNWZ( <Hermann-Mauguin-symbol> )

returns a representative, <S>, of the space group type specified by 
the arguments. <S> is returned in the form of an `AffineCrystGroupOnLeft',
which acts from the left on column vectors (see also the description
of the `DisplaySpaceGroupGenerators' function above). The package
{\Cryst} provides methods for the computation with space groups.

\beginexample
gap> S := SpaceGroupOnLeftBBNWZ( "P61" );
SpaceGroupOnLeftBBNWZ( 3, 6, 1, 1, 4 )
gap> for s in GeneratorsOfGroup( S ) do
>  Print( "\n" ); PrintArray( s ); od; Print( "\n" );

[ [   -1,    0,    0,    0 ],
  [    0,   -1,    0,    0 ],
  [    0,    0,    1,  1/2 ],
  [    0,    0,    0,    1 ] ]

[ [    0,   -1,    0,    0 ],
  [    1,   -1,    0,    0 ],
  [    0,    0,    1,  1/3 ],
  [    0,    0,    0,    1 ] ]

[ [  1,  0,  0,  1 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  1 ],
  [  0,  0,  1,  0 ],
  [  0,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  1 ],
  [  0,  0,  0,  1 ] ]

gap> CrystCatRecord( S ).parameters;
[ 3, 6, 1, 1, 4 ]
\endexample

The resulting group has an attribute `CrystCatRecord', whose component
`parameters' specifies the given space-group type.

\>SpaceGroupOnRightBBNWZ( <dim>, <system>, <q-class>, <z-class>, <sg-type> )
\>SpaceGroupOnRightBBNWZ( <dim>, <IT-number> )
\>SpaceGroupOnRightBBNWZ( <Hermann-Mauguin-symbol> )
\>SpaceGroupOnRightBBNWZ( <S> )

returns a representative, <T>, of the space group type specified by 
the arguments. <T> is returned in the form of an `AffineCrystGroupOnRight',
which acts from the right on row vectors. The generators of <T> are the 
transposed generators (in the same order) of the corresponding 
`SpaceGroupOnLeftBBNWZ', <S>, specified by the same arguments.
The space group <S> is also accepted as argument. The package
{\Cryst} provides methods for the computation with space groups.

\beginexample
gap> T := SpaceGroupOnRightBBNWZ( S );
SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 )
gap> for m in GeneratorsOfGroup( T ) do
>  Print( "\n" ); PrintArray( m ); od; Print( "\n" );
    
[ [   -1,    0,    0,    0 ],
  [    0,   -1,    0,    0 ],
  [    0,    0,    1,    0 ],
  [    0,    0,  1/2,    1 ] ]

[ [    0,    1,    0,    0 ],
  [   -1,   -1,    0,    0 ],
  [    0,    0,    1,    0 ],
  [    0,    0,  1/3,    1 ] ]

[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  1,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  1,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  0,  1,  1 ] ]

\endexample

\>SpaceGroupBBNWZ( <dim>, <system>, <q-class>, <z-class>, <sg-type> )
\>SpaceGroupBBNWZ( <dim>, <IT-number> )
\>SpaceGroupBBNWZ( <Hermann-Mauguin-symbol> )

calls either `SpaceGroupOnLeftBBNWZ' or `SpaceGroupOnRightBBNWZ'
with the same arguments, depending on the value of the variable
`CrystGroupDefaultAction'.

\>FpGroupSpaceGroupBBNWZ( <S> )

returns a finitely presented group <G>, say, which is isomorphic to
<S>, where <S> is expected to be a space group from the BBNWZ
catalog (acting from the left or from the right).  It is chosen such
that there is an isomorphism from <G> to <S> which maps each generator
of <G> onto the corresponding generator of <S>.  This means, in
particular, that the matrix generators of <S> satisfy the relators of
<G>. If the factor group of <S> by its translation normal subgroup is
solvable, then the presentation returned is a polycyclic power
commutator presentation.

\beginexample
gap> G := FpGroupSpaceGroupBBNWZ( S );    
FpGroupSpaceGroupOnLeftBBNWZ( 3, 6, 1, 1, 4 )
gap> for rel in RelatorsOfFpGroup( G ) do Print( rel, "\n" ); od;
g1^2*g5^-1
g2^3*g5^-1
g2^-1*g1^-1*g2*g1
g3^-1*g1^-1*g3*g1*g3^2
g3^-1*g2^-1*g3*g2*g4*g3^2
g4^-1*g1^-1*g4*g1*g4^2
g4^-1*g2^-1*g4*g2*g4*g3^-1
g4^-1*g3^-1*g4*g3
g5^-1*g1^-1*g5*g1
g5^-1*g2^-1*g5*g2
g5^-1*g3^-1*g5*g3
g5^-1*g4^-1*g5*g4
gap> # Verify that the matrix generators of S satisfy the relators of G.
gap> ForAll( RelatorsOfFpGroup( G ), rel ->
>  MappedWord( rel, GeneratorsOfGroup(G), GeneratorsOfGroup(S) ) = One(S) );
true
\endexample





