%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Accessing the data}
When this package is loaded, then the groups of order $3^8$ and $p^7$ for
primes $p > 11$ are additionally available via the SmallGroups library. As
a result, all groups of order $p^n$ with $p=2$ and $n \leq 9$ and $p=3$ and
$n \leq 8$ and $p$ an arbitrary prime and $n \leq 7$ are then available via
the small groups library. The corresponding information can be obtained via
\>SmallGroup(size, number)
\>NumberSmallGroups(size)
\>SmallGroupsInformation(size)
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.
WARNING: The user should be aware that there are there are 1,396,077 groups
of order $3^8$, 1,600,573 groups of order $13^{7}$, and 5,546,909 groups
of order $17^7$. For general $p$ the number of groups of order $p^7$ is
a PORC polynomial in $p$ with leading term $3p^5$. Furthermore, as the prime
$p$ increases, the time taken to generate a complete list of the groups of
order $p^7$ grows rapidly. Experimentally the time seems to be proportional
to $p^{6.2}$. For $p=13$ it takes several hours to generate the complete
list. For primes $p\leq 11$ the groups are precomputed, and their SmallGroup
codes are stored in the SmallGroups database. For primes $p>11$ 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 $13^7$ takes over 11 gb of memory.
\beginexample
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 \endexample
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