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<html><head><title>[sglppow] 3 The organization of the data</title></head>
<body text="#000000" bgcolor="#ffffff">
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<h1>3 The organization of the data</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">The $p$-group generation algorithm</a>
<li> <A HREF="CHAP003.htm#SECT002">The groups of order 6561</a>
<li> <A HREF="CHAP003.htm#SECT003">The groups with order the seventh power of a prime</a>
</ol>

We include some brief comments on the organization of the data. As a
preliminary step we briefly recall the p-group generation algorithm.


<h2><a name="SECT001">3.1 The $p$-group generation algorithm</a></h2>

The p-group generation algorithm was developed and implemented by Eamonn
O'Brien, and we refer the reader to OBrien90 for a detailed
description of the algorithm.

For a brief overview, let P be a p-group. The algorithm uses the lower
p-central series, defined recursively by λ1(P)=P and
λi+1(P)=[λi(P),P]λi(P)p for i ≥ 1.
The p-class of P is the length of this series. Each p-group P,
apart from the elementary abelian ones, is an immediate descendant of
the quotient P/R where R is the last non-trivial term of the lower
p-central series of P.

Thus all the groups with order 38, except the elementary abelian one,
are immediate descendants of groups with order 3k for some k smaller
than 8. All of
the immediate descendants of a p-group Q are quotients of a certain
extension of Q; the isomorphism problem for these descendants is equivalent
to the problem of determining orbits of certain subgroups of this extension
under an action of the automorphism group of Q. Not all p-groups have
immediate descendants, those that do are called capable, and those which
do not are called terminal.

O'Brien and Vaughan-Lee's classification of the groups of order p7
in <a href="biblio.htm#OVL05"><cite>OVL05</cite></a> is based on a classification of the nilpotent Lie rings of
order p7, and the groups of order p7 are obtained from the Lie
rings using the Baker-Campbell-Hausdorff formula. O'Brien and Vaughan-Lee
classified the nilpotent Lie rings of order p7 using the nilpotent
Lie ring generation algorithm, which is a direct analogue of the p-group
generation algorithm.

Thus the databases of nilpotent Lie rings of order
p7 and of the groups of order 38 are organized according to these
algorithms: the immediate descendants of order p7 of each nilpotent
Lie ring of order less than p7 are grouped together in the database of
nilpotent Lie rings, and the immediate descendants of order 38 of each
group of order less than 38 are grouped together in the database of
groups of order 38.


<h2><a name="SECT002">3.2 The groups of order 6561</a></h2>

The database of groups of order 38 is organized according to rank and
p-class. Here rank is the rank of the Frattini quotient, i.e. the size of
a minimal generating set, and p-class is as defined in the previous
section. The following table gives the number of groups of order 38 of
each rank and p-class, with the (i,j) entry corresponding to rank i
and p-class j.

<pre>
 | 1 2 3 4 5 6 7 8
--|-------------------------------------------------
1 | 0 0 0 0 0 0 0 1
2 | 0 0 58 486 1343 330 9 0
3 | 0 4 216747 40521 2163 24 0 0
4 | 0 23361 494666 22343 51 0 0 0
5 | 0 578478 14796 80 0 0 0 0
6 | 0 566 39 0 0 0 0 0
7 | 0 10 0 0 0 0 0 0
8 | 1 0 0 0 0 0 0 0
</pre>

In the list of all groups of order 38 the first group is the cyclic
group, then the 2-generator groups follow in order of increasing p-class
and so on. The above table can thus be used to find the range of numbers
of groups with a given rank and p-class.

As mentioned above, the database is organized according to the p-group
generation algorithm. For example, the 9 groups of rank 2, p-class 7, and
order 38 are numbered from 2219--2227. The groups numbered 2219 and
2220 are descendants of SmallGroup(37,384), and the groups numbered
2221--2227 are descendants of SmallGroup(37,386). Similarly, the 24
groups of rank 3, p-class 6, and order 38 are numbered from
261663--261686. The first four of these groups are descendants of
SmallGroup(37,5841), the next 17 are descendants of
SmallGroup(37,5844), and the last 4 are descendants of
SmallGroup(37,5849).


<h2><a name="SECT003">3.3 The groups with order the seventh power of a prime</a></h2>

The groups of order p7 for prime p > 11 are obtained from the LiePRing
database of nilpotent Lie rings of order p7 using Willem de Graaf's
implementation of the Baker-Campbell-Hausdorff formula. The LiePRing
database is organized according to the output from the nilpotent Lie ring
generation algorithm. For any given p the first Lie ring in the database
is the cyclic Lie ring of order p7. Next come the two generator Lie
rings, then the three generator Lie rings, and so on, ending with the six
generator Lie rings, and then finally the elementary abelian Lie ring of
rank 7. The first four of the two generator nilpotent Lie rings of order
p7 are immediate descendants of the Lie ring
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 〈a,b | pb, class 3〉</td></tr></table></td></tr></table>
of order p4. The next p2+8p+25 are immediate descendants of the
Lie ring
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 〈a,b | baa,bab,pba, class 3〉</td></tr></table></td></tr></table>
of order p5, and the next p+6+(p2+3p+10) gcd(p−1,3) are immediate
descendants of
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 〈a,b | babb,pa,pb, class 4〉·</td></tr></table></td></tr></table>
The nine rank 6 Lie rings in the database are the rank 6, p-class 2
Lie rings. These are the immediate descendants of the elementary abelian
Lie ring of rank 6.

There is a complete list of presentations for the nilpotent Lie rings of
order pk for k ≤ 7 valid for all p > 3 in the document p567.pdf
supplied with the documentation for the LiePRing package. The presentations
are grouped as described above, with each group of presentations giving the
immediate descendants of a Lie ring of smaller order.

In a few cases the descendants of a parametrized family of Lie rings are
grouped together. For example there is a family of p(p−1) distinct Lie
rings with presentations of the form
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 〈a,b,c | ca−baa, cb, pa−λbaa−μbab, pb+νbaa+ξbab, pc, class 3〉</td></tr></table></td></tr></table>
with λ,μ,ν,ξ ≠ 0. Most of these algebras are terminal,
but [5/2]p−[9/2]+[1/2]gcd(p−1,4) of them are capable,
and they have a total of
[1/2]p3+2p2−5p+[1/2]+[(p)/2]gcd(p−1,4)
descendants of order p7 and p-class 4. These
descendants have presentations

 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 〈a,b,c | ca−baa, cb, pa−baa−μbab−ybaaa, pb+νbaa+bab−zbaaa, pc−tbaaa, class 4〉, </td></tr></table></td></tr></table>

 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 〈a,b,c | ca−baa, cb, pa−baa−μbab−ybaaa, pb+νbaa+μνbab−zbaaa, pc−tbaaa, class 4〉</td></tr></table></td></tr></table>

for various choices of the parameters μ,ν,y,z,t. For any given value
of p the [1/2]p3+2p2−5p+[1/2]+[(p)/2]gcd(p−1,4)
distinct Lie algebras with presentations of this form are grouped together,
with consecutive numbering.

There is no easy way to determine the numbering of (say) the three generator
Lie rings of p-class 4 since the numbers depend on p in a very
complicated way, and generally there is no easy efficient way of searching
the database for a group with particular properties. In view of the numbers
of groups of order p7 and the time needed to generate a complete list,
this means that the database will be of limited use for most people. A user
who wants to access a particular batch of descendants as described in
p567.pdf is advised to use the LiePRing package directly, as this package
also has an option to obtain the corresponding groups via Willem de Graaf's
implementation of the Baker-Campbell-Hausdorff formula. On the other hand,
there are only

 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 2p3+13p2+64p+145+(p2+10p+56) gcd(p−1,3)+(4p+28) gcd(p−1,4) </td></tr></table></td></tr></table>
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> +(2p+12) gcd(p−1,5)+ gcd(p−1,7)+4 gcd(p−1,8)+ gcd(p−1,9) </td></tr></table></td></tr></table>
two generator groups of order p7 for p > 5 and a complete list of them
can be generated quite quickly for moderate values of p. The table below
gives the number of d generator groups of order p7 valid for all
p > 5, and the user can use the table to compute the range of numbers needed
to access the d generator groups of order p7 for any given p.


<dl compact>
<dt><var>rank 1</var> <dd> 1

<var>rank 2</var>
<dt> <dd> 2p3+13p2+64p+145+(p2+10p+56)gcd (p−1,3) +(4p+28)gcd (p−1,4)
<dt> <dd> +(2p+12)gcd (p−1,5)+gcd (p−1,7)+4gcd (p−1,8)+gcd (p−1,9)

<var>rank 3</var>
<dt> <dd> 2p5+9p4+29p3+99p2+380p+1100+(3p2+28p+189)gcd (p−1,3)
<dt> <dd> +(p2+13p+84)gcd (p−1,4)+(p+17)gcd (p−1,5)+gcd (p−1,8)+3gcd (p−1,7)

<var>rank 4</var>
<dt> <dd> p5+3p4+13p3+57p2+248p+1044+(6p+46)gcd (p−1,3)
<dt> <dd> +(2p+23)gcd(p−1,4)+2gcd(p−1,5)

<dt><var>rank 5</var> <dd> p2+15p+155

<dt><var>rank 6</var> <dd> 9

<dt><var>rank 7</var> <dd> 1
</dl>

The total number of groups of order p7 for all p > 5 is given by

 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> 3p5+12p4+44p3+170p2+707p+2455 </td></tr></table></td></tr></table>
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> +(4p2+44p+291)gcd (p−1,3)+(p2+19p+135)gcd (p−1,4) </td></tr></table></td></tr></table>
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> +(3p+31)gcd (p−1,5)+4gcd (p−1,7)+5gcd (p−1,8)+gcd (p−1,9)·</td></tr></table></td></tr></table>


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