When this package is loaded, the groups of order \(3^8\) and \(p^7\) for primes \(p > 11\) become additionally available via the Small Groups Library. As a result, all groups of order \(p^n\) with \(p=2\) and \(n \leq 9\), with \(p=3\) and \(n \leq 8\), and with \(p\) an arbitrary prime and \(n \leq 7\) are then available via the Small Groups Library. The corresponding information can be obtained via the following functions.
‣ SmallGroup( size, number ) | ( function ) |
‣ NumberSmallGroups( size ) | ( function ) |
‣ SmallGroupsInformation( size ) | ( function ) |
See smallgrp: The Small Groups Library 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 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 Small Groups Library. 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.
gap> NumberSmallGroups(3^8); 1396077 gap> SmallGroup(3^8, 1000000); <pc group of size 6561 with 8 generators>
The groups of order \(p^7\) for \(p>11\) are only available if the LiePRing and LieRing packages are loaded. They are constructed in batches, so the first request for such a group reports what it is doing and may take a while:
gap> NumberSmallGroups(17^7); 5546909 gap> NumberSmallGroups(101^7); 32826263845 gap> SmallGroup(17^7, 5000); constructing a batch of 1156 groups ... this may take a while <pc group of size 410338673 with 7 generators>
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