Database of groups of prime-power order for some prime-powers

Version 2.1

Released 2018-03-08

This project is maintained by Michael Vaughan-Lee, Bettina Eick

README file for SglPPow SglPPow is an extension to the GAP Small Groups Library. Currently the Small Groups Library gives access to the following groups: Those of order at most 2000 except 1024 (423,164,062 groups); Those of cubefree order at most 50,000 (395,703 groups); Those of order p^7 for the primes p = 3,5,7,11 (907,489 groups); Those of order p^n for n <= 6 and all primes p; Those of order q^n * p where q^n divides 28, 36, 55 or 74 and p is an arbitrary prime not equal to q; Those of squarefree order; Those whose order factorises into at most 3 primes. This package gives access to the groups of order p^7 for primes p > 11, and to the groups of order 3^8. The Database of groups of order 3^8 has been determined by Michael Vaughan-Lee. Access to the groups of order p^7 for primes p > 11 is via Bettina Eick and Michael Vaughan-Lee's LiePRing package which is based on Eamonn O'Brien and Michael Vaughan-Lee's classification of the nilpotent Lie rings of order p^7. The package can be downloaded as a .tar.gz file from https://gap-packages.github.io/sglppow/ Then tar -zxvf slgppow.tar produces a directory sglppow. This should be moved into the pkg directory of a GAP installation. The package is set up so that after loading it into GAP with LoadPackage("sglppow"), the groups can be obtained via the command SmallGroup( size, nr ) You can also obtain the number of groups of a given order with the command NumberSmallGroups(size) Thus the package does not install any new functionality in GAP, it only extends the available SmallGroups library. To access the groups of order p^7 for p > 11 you will also need to install the LiePRing package and the LieRing package due to Willem de Graaf and Serena Cicalo. These packages are automatically loaded when SglPPow is loaded. WARNING: 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 of order 3p^5. Furthermore as p increases, the time taken to generate a complete list of the groups of order p^7 grows rapidly. Experimentally the time taken seems to be proportional to p^{6.2}. For p=13 it takes several hours to generate the complete list. For p <= 11 the groups are precomputed, and their SmallGroup codes are stored in the SmallGroup database. But for p > 11 the Lie rings have to be generated from a list of 4773 parametrized presentations in the LiePRing database, and then converted into groups using the Baker-Campbell-Hausdorff formula. Further, it takes over 11 gb of memory to store a complete list of power-commutator presentations for all groups of order 13^7. Hence most users will want to avoid generating complete lists of the groups!