<html><head><title>[sglppow] 2 Accessing the data</title></head>
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<h1>2 Accessing the data</h1><p>
<p>
When this package is loaded, then the groups of order 3<sup>8</sup> and <i>p</i><sup>7</sup> for
primes <i>p</i> > 11 are additionally available via the SmallGroups library. As
a result, all groups of order <i>p</i><sup><i>n</i></sup> with <i>p</i>=2 and <i>n</i> ≤ 9 and <i>p</i>=3 and
<i>n</i> ≤ 8 and <i>p</i> an arbitrary prime and <i>n</i> ≤ 7 are then available via
the small groups library. The corresponding information can be obtained via
<p>
<a name = ""></a>
<li><code>SmallGroup(size, number)</code>
<p>
<a name = ""></a>
<li><code>NumberSmallGroups(size)</code>
<p>
<a name = ""></a>
<li><code>SmallGroupsInformation(size)</code>
<p>
See Section 50.7 in the GAP manual for background on these functions. Note
that there is no IdGroup function available for this extension of the small
groups library.
<p>
WARNING: The user should be aware that there are there are 1,396,077 groups
of order 3<sup>8</sup>, 1,600,573 groups of order 13<sup>7</sup>, and 5,546,909 groups
of order 17<sup>7</sup>. For general <i>p</i> the number of groups of order <i>p</i><sup>7</sup> is
a PORC polynomial in <i>p</i> with leading term 3<i>p</i><sup>5</sup>. Furthermore, as the prime
<i>p</i> increases, the time taken to generate a complete list of the groups of
order <i>p</i><sup>7</sup> grows rapidly. Experimentally the time seems to be proportional
to <i>p</i><sup>6·2</sup>. For <i>p</i>=13 it takes several hours to generate the complete
list. For primes <i>p</i> ≤ 11 the groups are precomputed, and their SmallGroup
codes are stored in the SmallGroups database. For primes <i>p</i> > 11 the Lie rings
have to be generated from 4773 parametrized presentations in the LiePRing
database, and then converted into groups using the Baker-Campbell-Hausdorff
formula. A complete list of power commutator presentations for the groups
of order 13<sup>7</sup> takes over 11 gb of memory.
<p>
<pre>
gap> NumberSmallGroups(3^8);
1396077
gap> SmallGroup(3^8, 1000000);
<pc group of size 6561 with 8 generators>
gap> NumberSmallGroups(17^7);
5546909
gap> SmallGroup(17^7, 5000);
constructing a batch of 1156 groups ... this may take a while
<pc group of size 410338673 with 7 generators>
gap> NumberSmallGroups(101^7);
32826263845
</pre>
<p>
<p>
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<address>sglppow manual<br>March 2024
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