When this package is loaded, then the groups of order 38 and p7 for primes p > 11 are additionally available via the SmallGroups library. As a result, all groups of order pn with p=2 and n ≤ 9 and p=3 and n ≤ 8 and p an arbitrary prime and n ≤ 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 38, 1,600,573 groups of order 137, and 5,546,909 groups of order 177. For general p the number of groups of order p7 is a PORC polynomial in p with leading term 3p5. Furthermore, as the prime p increases, the time taken to generate a complete list of the groups of order p7 grows rapidly. Experimentally the time seems to be proportional to p6·2. 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 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 137 takes over 11 gb of memory.
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
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