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 ≤ 9, with p=3 and n ≤ 8, and with p an arbitrary prime and n ≤ 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≤ 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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