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#W  README             The  Small Groups Library           Hans Ulrich Besche
##                                                               Bettina Eick
##                                                             Eamonn O'Brien
##
##  This file contains information about the Small Groups Library
##
"@(#)$Id: README,v 1.3 2001/09/07 05:38:06 gap Exp $";

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# Authors
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The Small Groups Library has been developed by

Hans Ulrich Besche
Lehrstuhl D fuer Mathematik
RWTH Aachen
Templergraben 64
52062 Aachen
Germany

e-mail: Hans-Ulrich.Besche@math.rwth-aachen.de
http://www.math.rwth-aachen.de/~Hans-Ulrich.Besche/


Bettina Eick
Fachbereich Mathematik und Informatik
Technische Universität Braunschweig
Pockelsstr. 14
38106 Braunschweig
Germany

e-mail: beick@tu-bs.de
http://www.tu-bs.de/~beick/

Eamonn A. O'Brien
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand 

e-mail: obrien@math.auckland.ac.nz
http://www.scitec.auckland.ac.nz/~obrien/

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# Introduction
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The Small Groups Library is a catalogue of groups of `small' order. 
Currently, (Jan. 2000) it contains the groups 

    * of order at most 2000 except 1024.
    * of order 5^5 and 7^4.
    * whose order factorises in at most 3 primes.
    * of order q^n * p for a prime-power q^n dividing 2^8, 3^6, 5^5 or 7^4 
      and an arbitrary prime p different to q.

The Small Groups Library provides access to these groups and a method to 
identify the catalogue number of a given group. Here we describe the organi-
sation of this catalogue. We include information on the determination of the 
groups, the encoding of the stored groups and the algorithms used for the 
identification routine. 

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# Available functions
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SmallGroup( order, number ) 
    returns the specified group.

AllSmallGroups( order, selection-args ) 
    returns all or certain groups of the given order. The optional argument
    list of selection-args can be used to specify certain properties to the 
    function and only the groups with these properties are returned then. 
    For some properties and some group orders the desired groups are pre-
    comuted and stored. Otherwise the desired groups are filtered out of the 
    catalogue. See the GAP 4 manual for some examples.
    Synonym: AllGroups

OneSmallGroup( order, selection-args )
    returns one group of the given order with the specified properties. 
    Synonym: OneGroup

SmallGroupsInformation( order ) 
    prints information on the groups of the given order. This includes 
    a description of the determination and sorting of the groups of the
    given order.

NumberSmallGroups( order )
    returns the number of groups of the given order.
    Synonym: NrSmallGroups

IdGroup( G )
    returns the catalogue number of the group G. Clearly, the groups of
    order |G| must be available in the library for this purpose. Moreover,
    the groups of orders 512 and 1536 are excluded.

IdsOfAllSmallGroups( order, selection-args )
    similar to AllSmallGroups, but returns the catalogue numbers of the
    desired groups only. This can be useful if the list of explicit groups 
    would need to much storage space to load them.
    Synonym: IdsOfAllGroups

UnloadSmallGroupsData()
    clears the workspace from loaded data. If some groups from the 
    catalogue are requested from GAP afterwards, then the data will 
    be loaded again automatically.

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# Layers
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The Small Groups Library is organised in 8 layers. Each layer contains the
groups of certain orders:

     Layer 1: the orders with at most 3 prime factors; that is, the orders
              of type p, p^2, pq, p^3, pq^2 and pqr.

     Layer 2: all remaining orders <= 1000 except 512 and 768.

     Layer 3: all remaining orders of type 2^n * p with n <= 8 and 
              p an arbitrary odd prime.

     Layer 4: the orders 7^4 and 5^5 and all remaining orders of type 
              q^n * p with q^n dividing 3^6, 5^5 or 7^4 and p a prime 
              different to q.

     Layer 5: all remaining orders <= 2000 except 512, 1024, 1152, 1536 
              and 1920.

     Layer 6: the orders 1152 and 1920.

     Layer 7: the order 512. 

     Layer 8: the order 1536. 

This setup has been chosen for two reasons. First, the organisation in 
layers permits to add new layers (and thus new group orders) without changes 
to the existing catalogue. Secondly, it is possible to install a part of the 
layers only. This might be interesting for computer-systems with few hard 
disk space available.

For each layer i there are two internal functions:
   `SMALL_AVAILABLE_FUNCS[i]( order )' and `ID_AVAILABLE_FUNCS[i]( order )'.
Both functions return fail, if the given order is not contained in this 
layer. Otherwise the functions return an information record on the layer 
and the order. This record contains the information needed to handle the 
groups of the specified order internally. If the layer i is not installed, 
then SMALL_AVAILABLE_FUNCS[i] and ID_AVAILABLE_FUNCS[i] are both unbound.

Moreover, there are two internal header functions:
       `SMALL_AVAILABLE( order )' and `ID_AVAILABLE( order )'. 
These two functions are used to loop over the layer functions and find the 
first layer containing the given order. If this layer is found, then an 
information record on the order is returned. If there is no layer installed
containing the groups of this order, then the functions return fail.

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# Files and Directories
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For each layer x there are usually two directories in small:
                   `smallx' and `idx'
which contain the groups and the identification routine. An exception to
this rule is the first layer; here all the code for the groups is contained 
in the files smlgp1.g and idgrp1.g in the main directory of small. Two more
exceptions are the layers 7 and 8 where there is no identitfication routine
available at the moment and thus there are no directories id7 and id8.

The other files in the small main directory are:

  * README           - this file
  * readsml.g        - for loading the small groups catalogue into GAP
  * small.gi/gd      - these are the files where the main functions for the
                       small groups catalogue are defined.
  * gap3cat.g        - identification of a small group in the Gap 3 
                       `SolvableGroup' catalogue
  * smlinfo.gi       - the code for the function `SmallGroupsInformation'

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# Layer 1 
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The groups whose order factorises in at most 3 primes have been classified 
by O. H"older, see [8]. An algorithm to find the isomorphism type of such 
groups was first implemented in Gap 3 by Frank Celler and Hans-Georg Esser. 

H"olders classification distinguishes the groups of the following order 
types. Let p, q and r are different primes with p < q < r.

  p                - 1 cyclic group
  p^2              - 1 cyclic and 1 elementary abelian group
  p^3              - 3 abelian groups and 2 extraspecial groups

  pq               - 1 cyclic group 
                     1 group of type q:p if q = 1 mod p

  p^2q             - 2 abelian groups
                     1 group of type q:p x p if q = 1 mod p
                     1 group of type A4 if p = 2 and q = 3
                     1 group of type (pxq).p if q = 1 mod p
                     1 group of type q:C(p^2) if q = 1 mod p^2 

  pq^2             - 2 abelian groups
                     1 group of type q:p x q if q = 1 mod p
                     1 group of type C(q^2):p if q = 1 mod p
                     1 group of type (qxq):p if p > 2 and q+1 = 0 mod p 
                     groups of type q:p + q:p
                       (1 group for p = 2 and (p+1)/2 groups otherwise)

  pqr              - 1 cyclic group
                     1 group of type q:p x r if q = 1 mod p
                     1 group of type r:p x q if r = 1 mod p
                     1 group of type r:q x p if r = 1 mod q
                     1 group of type r:(pxq) if r = 1 mod (p*q)
                     p-1 groups of type q:p + r:p 

The groups in this layer are stored `generically'; that is, using functions 
instead of explicit presentations. The identification routine follows the 
classification as well.

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# Layer 2
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DETERMINATION:
The nilpotent groups of this layer have been constructed using p-group 
generation, see [10] and [12]. The 2- and 3-groups of this layer have also 
been avaliable in the Two- and ThreeGroup library of Gap 3, see [9] and [11]
also. The non-nilpotent groups in this layer have been determined using the 
Frattini extension method, the cyclic split extension method and cyclic 
extension for the non-soluble groups, see [1] and [2]. 

STORING:
The solvable groups in this layer are stored by a single long integer 
describing a pc presentation of the group (see `PcGroupCode' or [1]). 
For p-groups the encoded pc presentations are standard presentations and
they are equal to the presentations known in the 2- and 3-groups libraries
of GAP 3. The non-solvable groups are stored by generating permutations.

SELECTION-ARGS:
For the selection functions the values of the following attributes are
precomputed and stored:

  for p-groups:
     IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and 
     FrattinifactorId 

  for non-p-groups:
     IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, 
     LGLength, FrattinifactorSize and FrattinifactorId 

IDENTIFICATION:
The identification routine uses invariants of groups to identify a given 
group. The invariants are organised as tree and stored in `ID_GROUP_TREE', 
which contains a leave for every group. In some cases this invariants tree 
is not sufficient to distinguish all groups. In this case the tree is used 
to reduce to a small number of possible groups and then a special random 
isomorphism test finds the correct group among the possible ones.

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# Layer 3
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DETERMINATION:
These groups have been determined using the generic variation of the cyclic 
split extension method and the Frattini extension method, see [3].

STORING:
* Nilpotent groups:
For each order in this layer we first list the nilpotent groups P x Q where
P is the cyclic group of order p and Q is a group of order 2^n. These groups
are sorted by the catalogue number of Q and the group Q is stored in layer 2.

* Groups with normal Sylow p-subgroup:
Then we consider the remaining groups with normal Sylow p-subgroup. These 
are split extensions of type P : Q. The isomorphism type of these groups 
depends on the isomorphism type of Q and the action homomorphism Q -> Aut(P). 
Let K be the kernel of this homomorphism with [Q : K] = 2^i for some i. We
encode the homomorphisms for fixed Q and fixed i as follows.

Consider a generator g of P and let g1, ..., gr be a generating set of Q
(we choose the first r pc generators of Q for r the rank of Q). Consider 
a homomorphism a : Q -> Aut(P) and let ai be the image of gi. Clearly, ai 
is of the form ai : g -> g^li with 0 <= li <= 2^i-1. Thus we describe the 
homomorphism a by the sequence l1, ..., lr.  We encode this sequence as
(2^i-1)-adic number of length r; that is, we encode this sequence as
l1 * q^(r-1) + l2 * q^(r-2) + ... + lr * q^0 where q = 2^i. Hence each
homomorphism a is encoded as integer and for fixed Q and i we write all
these integers into a list.

A special case occurs if there are two homomorphisms a and b such that
ai = bi^j for some integer j; that is, the images of a are powers of the
images of b. In this case a is stored directly after b and we only store 
the integer -j instead of encoding for the sequence l1, ..., lr for a.

Now we consider the case that for certain Q and i we have a list L consisting
of non-negative integers only; that is, the special power-case did not occur 
for this group Q and index i. In some cases we encode such a list L further. 
Recall that the integers in L correspond to different homomorphisms. Thus the 
integers are all different and we may sort them in increasing order. Then we 
can store this sorted list as binary number; that is, the list c1, ..., cs is 
encoded as 2^(c1-1) + ...  + 2^(cs-1). This is only useful, if the list L 
does not contain large numbers.

Thus for Q and i we have either a list or an integer stored to encode the 
corresponding homorphisms. For each index i we write all lists or integers 
into a list such that at position Id(Q) the corresponding list or integer
to Q and i is found.

For i = 1 this list has no holes, since there exists at least one group 
P : Q with K of index 2 for each group Q. Often the entry at position j 
in this list is equal to the entry at position j-1. In this case we delete 
the entry at position j and thus create a hole in the list. 

For i > 1 there might be holes in the list if there is no group P : Q with 
K of index 2^i. Hence the previous idea cannot be applied for i > 1. However, 
in all the lists we have equal entries in various places. If this is the case,
then we substitute an entry in a list by a negative integer -j meaning that 
the entry at this position is equal to the entry at position j.

Finally, if the list for some index i has only few entries left, then we 
substitute the list by a record containing two lists: first the non-empty 
positions and secondly their entries.

* Groups with normal Sylow 2-subgroup:
Next we consider the remaining groups of type Q : P. These exist for a few
primes q only. These groups are described by operation homomorphisms of the 
type P -> Aut(Q). Here we store the catalogue number of Q and a long integer 
for each such homomorphism a : P -> Aut(Q). Consider a generator g of P. 
Clearly, we just need to store the image of g for each homomorphism a. The 
image g^a is an automorphism of Q and thus we store the images under g^a
of a fixed generating set of Q. For this purpose we consider a canonical 
numbering of the elements of Q and store the generator images by storing 
their number in the element list. The sequence of these numbers are encoded
as (|Q|+1)-adic number; that is, the sequence e1, ..., er is stored as 
e1 * q^(r-1) + e2 * q^(r-2) + ... + er where q = |Q| + 1.

* Groups without normal Sylow subgroup:
Now there are the groups of order 2^n * p without any normal Sylow subgroup
left. These exist for very few primes p only. They have been computed as in 
layer 2 using the Frattini extension method. The are stored as described in
layer 2 using one long integer for each group.

IDENTIFICATION:
The identification of nilpotent groups P x Q relies on the identification of
Q only and this is done as in layer 2. Similarly, the groups without normal 
Sylow subgroup are identified as described in layer 2.

For the groups Q : P with normal Sylow q-subgroup we first determine the prime 
p and the isomorphism type of Q. If this is not sufficient, then we use the 
methods as described in layer 2. (Recall that there are only finitely many 
primes p possible here.)

For the groups P : Q with normal Sylow p-subgroup we first determine the 
prime p and the isomorphism type of Q as well. Moreover, we compute the 
centralizer K of P in Q and consider its index [Q : K] = 2^i. 

In a large number of cases the catalogue numbers Id(Q) and Id(K) are 
sufficient to determine the group P:Q uniquely. Moreover, this feature
is independent of p. We can recognise these cases using the tree of
invariants: if we can determine P:Q uniquely with Id(Q) and Id(K) only,
then there is no node corresponding to this case in the tree.

If there is a node in the tree, then Id(Q) and Id(K) are not sufficient to 
determine the group uniquely. In this case we apply methods similar to 
layer 2; that is, we use invariants. Since there are infinitely many primes 
p which might turn up here, we need an additional idea to use the invariant 
tree for this purpose. In the invariant tree we have invariants stored for 
groups of type R : Q of order 2^n * r for primes r in 3, 5, 17 and 97. Now 
we first choose the smallest prime r such that r is congruent 1 mod [Q : K]. 
The cases that [Q : K] = 2^i for i = 6, 7 or 8 cannot occur here, because
in this cases Id(Q) and Id(K) are allways sufficient to determine the group.

Then we use that the groups P : Q behave essentially similar to the groups 
R : Q for the chosen r. Thus we can construct a group R : Q corresponding
to P : Q and identify R : Q instead of P : Q.

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# Layer 4
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DETERMINATION:
The groups of order 2401 = 7^4 and 3125 = 5^5 have been computed using 
p-group generation as described in layer 2 and they are stored and handled 
similarly. The other groups in this layer have been determined as outlined 
in layer 3 and, again, they are organised similarly. 

STORING:
There is a difference to layer 3 in the compression of the group data; that 
is, the groups in this layer are not compressed as efficiently as the groups 
in layer 3. We consider the groups of type P : Q for P the cyclic group of 
order p and Q a group of order q^n. These groups are determined by Q and 
an operation homomorphism Q -> Aut(P). As in layer 3 we first compute
one integer for each operation homomorphism and then proceed to compress
the lists of integers for each Q and each i where q^i is the index of the
centralizer of P in Q. Different to layer 3 we do not compress the lists of
non-negative integers to a single integer and we not reduce the list for
i = 1 further, since in both cases the obtained compression is very small.

IDENTIFICATION:
The group identification is essentially similar to the procedure described
in layer 3. However, for the groups with normal Sylow p-subgroup we use
a special random isomorphism test additionally to identify groups. It is
a `special' version of the general algorithm: in the general algorithm we
use certain sets to choose generating sets from. In this special version
we restrict these sets suitably. (We use non-trivial elements of P and 
certain elements of Q.) This is necessary, since the groups which appear
here can be to large to compute explicitly the elements sorted in conjugacy
classes.

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# Layer 5
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This layer is similar to layer 2.

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# Layer 6
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DETERMINATION:
The nilpotent groups in this layer are determined as direct products of
p-groups. Most of the groups of orders 1152 and 1920 have a normal Sylow
3-subgroup or Hall (3,5)-subgroup, resp. They are constructed using the 
coprime split extension method, see [5]. The remaining groups of these or-
ders (without normal 2-complement) have been determined using the Frattini 
extension method and the cyclic split extension method, see [1] also.

STORING:
Only the non-nilpotent groups with normal Sylow 3-subgroup or Hall 
(3,5)-subgroup are stored in a special way. The remaining groups are 
organised similar to layer 2. 

For the groups of type P : Q where P is the normal Sylow 3-subgroup
or the Hall (3,5)-subgroup and Q is a group of order 2^7 we split a 
pc presentation in two parts. The first part are the relators of Q. 
These are stored within the groups of order 2^7 and thus can be 
considered as known. The second part of the relators determine P and 
the operation of Q on P. These second parts are stored as long integers. 
If we consider these second parts for all the groups, then we find that 
they are highly redundant. There are only 921 (1722) different long 
integers for the groups of order 1152 and 1920, resp. Thus we store 
these two sets of long integers only.

Then for each group Q of order 2^7 we store for each extension P : Q
the position of the corresponding long integer. Hence we obtain a list
of integers for Q. Again we note that the lists of integers obtained
for all the groups Q of order 2^7 are highly redundant. There are only 
1298 (722) different lists for the groups of order 1152 (1920). Thus
we can compress the lists using the same idea again.

IDENTIFICATION:
The identification of groups of these orders is similar to layer 2.

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# Layer 7
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DETERMINATION:
The groups of order 512 have first been enumerated, see [6] and [7]. Later
they have been determined explicitly using p-group generation, see [10] and
[5]. There are 10494213 groups of this order.

STORING:
We introduce a special method to store such groups. Consider a pc presen-
tation of a given group of order 512. If we concatenate the exponent vectors
of the right hand sides of such a presentation, then we obtain a bit vector 
of length 120. This can be used as encoding of a group of order 512. We 
compress the list of such encodings further using the following method. 

First we split the list of all groups of order 512 in sublists containing 
1000 groups each. Let L be such a sublist. We suppose that L consists of 
1000 bitlists of length 120 and we describe a method to encode this list
of bitlists in a single vector V using an alphabet of 83 symbols. (81
symbols for the encoding and 2 special symbols as signs.)

First we find those indices i in the range 1 to 120 such that for all 
lists in L the entries at position i are fixed. Thus we note for each of
the 120 entries whether the entries are all 0, all 1 or variable in L.
There are 3^120 combinations possible which we store in a vector of length
30 using 81 symbols. This vector of length 30 is the initial segment of V.
Now we can reduce to consider the variable positions only and thus work 
with bitlists of shorter length.

We split the shorter bitlists in blocks such that the bitlists in each block 
have at most 6 variable entries. For this purpose we start at the beginning
of L and add bitlists to the current block until the next bitlist would lead
to more than 6 variable entries for this block. Each block can contain 1 up 
to 64 bitlists. Now we determine a `headcode' and a `tailcode' to describe 
the block. The head and tail codes are then concatenated to V. The head 
contains an encoding of the variable bits for this block as above. The tail 
then describes the vectors in the variable bits. Since they are at most 64 
possible combinations arising here, we need only one letter in the 81-alpha-
bet to describe a single vector of variable bits.

Finally, we note that some of the tailcodes arise for many blocks. These
tailcodes are stored in a special list. If such a tailcode is occuring
in V, then only the reference to the special list is stored instead of 
the full tailcode. This special list contains 1890 different tailcodes, 
2 letters in the alphabet are sufficient to refer a tailcode.

To distinguish references and actual tailcodes we use an special sign. Each
new head beginns with a special symbol in V. There are two special symbols
available for this purpose. One is used, if the corresponding tailcode is 
referenced, and the other, if it is an ordinary tailcode.

This compression allows to store the groups of order 512 in 4.4 mb.

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# Layer 8
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DETERMINATION:
The 408641062 groups of order 1536 have been determined using the cyclic
split extension method and the improved version of the Frattini extension
method, see [1] and [3]. Note that the number of groups of this order is
too large to be an ordinary GAP integer. Thus it is not possible to loop
over these groups with a for-loop in GAP.

STORING:
We store these groups in a special way as well and we obtain a compression
to 6.1 mb. The groups are stored using references to the groups of order 
512 of layer 7.

The nilpotent groups C3 x Q for Q a group of order 512 are based on the 
groups of order 512. 

The major part of the groups are the 398032384 non-nilpotent groups with 
normal Sylow 3-subgroup. The compression of these groups follows the one 
outlined in layer 3. As described there for each group Q of order 512 we 
compute an integer which determines all groups of type C3 : Q. There are 
only 15249 different integers of this type. In a first compression we 
create a list of references to the different integers.

Now we note that a group often has the same reference as the previous
one. Thus we split the groups of order 512 in sets of 100 000 groups and
create a record for each set. The record contains two lists which
are used to recall the blocks in a set having the same reference. 
Thus the first list contains the length of the blocks and the second
list contains the corresponding references. 

Again, in both lists certain patters occur often. In the first list
the length `1' appears often. Thus the `1' is not stored explicitly,
but a blank in this lists has to be considered as `1'. In the second
lists often an entry is the same as the one 2 places bevor. Thus these
entries are deleted as well and blanks can be reconstructed by this
rule. 

Additionally we store the number of resulting groups for each integer 
to speed up the method to find the group with a given catalogue number. 

The 18028 groups with normal Sylow 2-subgroup are stored similarly to 
layer 3. As described for layer 3 we determine a long integer for each
homomorphism Q -> Aut(P) for Q of order 512 and P the cyclic group of
order 3. The occuring integers are highly redundant - there are 6774 
different ones. Thus we use a list of references again.

The 96437 groups without normal Sylow subgroup are determined using the
Frattini extension method. As described in layer 2 they can be stored
by encoding the relators of a pc presentation as long integer. However, 
the relators corresponding to the Frattini factors of the groups are
known. Thus it is sufficient to store the relators of the Frattini factors
once for all groups with this factor and encode the remaining relators only. 
Note that there are only 16 different Frattini factors and only 12 of them 
have order less than 1536. The groups are stored in lists corresponding 
to the 12 Frattini factors.

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# Concluding remarks
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The compression of group data for layer 2 has been developed long ago. A
first version has been used in the p-group generation algorithm and thus
to encode the 2- and 3-groups library of Gap 3. This compression is des-
cribed in [1].

The remaining compression methods for layers 3 - 8 have been introduced by
Hans Ulrich Besche specially for this library. They are designed to store
the groups in as few space as possible. For different layers there are 
different methods, since certain approaches had been successful for certain 
groups but not for others. Moreover, already published layers have not been
changed again and thus useful concepts sometimes have not been applied to
earlier layers.

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# References
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#############################################################################

[1] Hans Ulrich Besche and Bettina Eick.
    The construction of finite groups.
    J. Symbolic Comput. 27, 387 -- 404 (1999).

[2] Hans Ulrich Besche and Bettina Eick.
    The groups of order at most 1000 except 512 and 768.
    J. Symbolic Comput. 27, 405 -- 413 (1999).

[3] Hans Ulrich Besche and Bettina Eick.
    The groups of order q^n * p.
    To appear in Comm. Algebra

[4] Hans Ulrich Besche and Bettina Eick.
    The GrpConst share package of GAP 4.

[5] Hans Ulrich Besche, Bettina Eick and E. A. O'Brien.
    A millenium project: constructing small groups.
    In preparation.

[6] Bettina Eick and E. A. O'Brien.
    The groups of order 512. 
    Proceedings of the `Abschlusstagung des DFG Schwerpunktes
    Algorithmische Algebra und Zahlentheorie', Springer (1998)

[7] Bettina Eick and E. A. O'Brien.
    Enumerating p-groups.
    Austal. Math. Soc. Ser. A, 67, 191 -- 205 (1999).

[8] Otto H"older.
    Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4.
    Math. Ann. 43, 301 -- 412 (1893).

[9] Rodney James, M. F. Newman and E. A. O'Brien.
    The groups of order 128.
    J. Algebra 129 (1), 136 -- 158 (1990)

[10] E. A. O'Brien.
     The p-group generation algorithm.
     J. Symbolic Comput. 9, 677 -- 698 (1990)

[11] E. A. O'Brien.
     The groups of order 256.
     J. Algebra 143 (1), 219 -- 235 (1991)

[12] E. A. O'Brien.
     The ANUPQ share package of Gap 3.

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# Bugs and problems
#
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Please report bugs/problems/feedback to the authors or to gap-trouble

Hans-Ulrich.Besche@math.rwth-aachen.de
beick@tu-bs.de
obrien@math.auckland.ac.nz
gap-trouble@dcs.st-and.ac.uk
