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1 Introduction

SglPPow is a package which extends the Small Groups Library. Currently the Small Groups Library gives access to the following groups:

1. Those of order at most 2000 except 1024 (423,164,062 groups);

2. Those of cubefree order at most 50,000 (395,703 groups);

3. Those of order \(p^{7}\) for the primes \(p=3,5,7,11\) (907,489 groups);

4. Those of order \(p^{n}\) for \(n\leq 6\) and all primes \(p\);

5. Those of order \(pq^{n}\) where \(q^{n}\) divides \(2^8\), \(3^6\), \(5^5\) or \(7^4\) and \(p\) is an arbitrary prime not equal to \(q\);

6. Those of squarefree order;

7. Those whose order factorizes 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 \(3^{8}\).

To access the groups of order \(p^{7}\) for primes \(p>11\) you need the packages LiePRing (by Michael Vaughan-Lee and Bettina Eick) and LieRing (by Willem de Graaf and Serena Cicalò).

The groups of order \(3^{8}\) have been determined by Michael Vaughan-Lee. The groups of order \(p^{7}\) for primes \(p>11\) are available via the database of the nilpotent Lie rings of order \(p^{k}\) for \(k\leq 7\) and primes \(p>3\) in the LiePRing package. These groups are obtained from the Lie rings using the implementation of the Baker-Campbell-Hausdorff formula in the LieRing package.

Acknowledgements: The authors thank Max Horn for help with the general framework of GAP programs used to extend the Small Groups Library.

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