[sglppow] 2 Accessing the data

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# 2 Accessing the data

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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