Constructing Groups of ‘Small’ Order: Recent Results and Open Problems

  • Bettina EickEmail author
  • Max Horn
  • Alexander Hulpke


We investigate the state of the art in the computational determination and enumeration of the groups of small order. This includes a survey of the available algorithms and a discussion of their recent improvements. We then show how these algorithms can be used to determine or enumerate the groups of order at most 20, 000 with few exceptions and we discuss the orders in this range which remain as challenging open problems.


Enumeration Determination Small groups Algorithms 

Subject Classifications

20D45 20E22 20-04 



We thank Eamonn O’Brien for comments on drafts of this work. The second author was supported by the DFG Schwerpunkt SPP 1489. The third author was supported by Simons Foundation Collaboration Grant 244502.


  1. 1.
    C. Archer, The extension problem and classification of nonsolvable groups. PhD Thesis, Université Libre de Bruxelles, 1998Google Scholar
  2. 2.
    H.U. Besche, B. Eick, Construction of finite groups. J. Symb. Comput. 27, 387–404 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    H.U. Besche, B. Eick, GrpConst - Construction of finite groups (1999). A refereed GAP 4 package, see [29]Google Scholar
  4. 4.
    H.U. Besche, B. Eick, The groups of order q n ⋅ p. Commun. Algebra 29(4), 1759–1772 (2001)Google Scholar
  5. 5.
    H.U. Besche, B. Eick, E.A. O’Brien, The groups of order at most 2000. Electron. Res. Announc. Am. Math. Soc. 7, 1–4 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    H.U. Besche, B. Eick, E.A. O’Brien, A millennium project: constructing small groups. Int. J. Algebra Comput. 12, 623–644 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    H.U. Besche, B. Eick, E. O’Brien, SmallGroups - a library of groups of small order (2005). A GAP 4 package; Webpage available at
  8. 8.
    S. Blackburn, P. Neumann, G. Venkataraman, Enumeration of Finite Groups (Cambridge University Press, Cambridge, 2007)CrossRefGoogle Scholar
  9. 9.
    J.J. Cannon, D.F. Holt, Automorphism group computation and isomorphism testing in finite groups. J. Symb. Comput. 35, 241–267 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Cayley, On the theory of groups, as depending on the symbolic equation θ n = 1. Philos. Mag. 4(7), 40–47 (1854)CrossRefGoogle Scholar
  11. 11.
    J. Conway, H. Dietrich, E. O’Brien, Counting groups: Gnus, Moas and other exotica. Math. Intell. 30, 6–15 (2008)CrossRefGoogle Scholar
  12. 12.
    B. Eick, M. Horn, The construction of finite solvable groups revisited. J. Algebra 408, 166–182 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Eick, E.A. O’Brien, Enumerating p-groups. J. Aust. Math. Soc. 67, 191–205 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    B. Eick, E. O’Brien, AutPGrp - computing the automorphism group of a p -group, Version 1.8 (2016). A refereed GAP 4 package, see [29]Google Scholar
  15. 15.
    B. Eick, C.R. Leedham-Green, E.A. O’Brien, Constructing automorphism groups of p-groups. Commun. Algebra 30, 2271–2295 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    D. Holt, W. Plesken, Perfect Groups (Clarendon Press, Oxford, 1989)zbMATHGoogle Scholar
  17. 17.
    M. Horn, B. Eick, GroupExt - Constructing finite groups (2013). A GAP 4 package, see [29]Google Scholar
  18. 18.
    M.F. Newman, E.A. O’Brien, M.R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime. J. Algebra 278, 383–401 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    E.A. O’Brien, The groups of order dividing 256. PhD thesis, Australian National University, Canberra, 1988Google Scholar
  20. 20.
    E. O’Brien, ANUPQ - the ANU p-Quotient algorithm (1990). Also available in Magma and as GAP packageGoogle Scholar
  21. 21.
    E.A. O’Brien, The p-group generation algorithm. J. Symb. Comput. 9, 677–698 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    E.A. O’Brien, M.R. Vaughan-Lee, The groups with order p 7 for odd prime p. J. Algebra 292(1), 243–258 (2005)Google Scholar
  23. 23.
    L. Pyber, Group enumeration and where it leads us, in European Congress of Mathematics, Volume II (Budapest, 1996), Progress in Mathematics, vol. 169 (Birkhäuser, Basel, 1998), pp. 187–199zbMATHGoogle Scholar
  24. 24.
    R. Schwingel, Two matrix group algorithms with applications to computing the automorphism group of a finite p-group. PhD Thesis, QMW, University of London, 2000Google Scholar
  25. 25.
    J.K. Senior, A.C. Lunn, Determination of the groups of orders 101–161, omitting order 128. Am. J. Math. 56(1–4), 328–338 (1934)MathSciNetCrossRefGoogle Scholar
  26. 26.
    J.K. Senior, A.C. Lunn, Determination of the groups of orders 162–215 omitting order 192. Am. J. Math. 57(2), 254–260 (1935)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M.J. Smith, Computing automorphisms of finite soluble groups. PhD thesis, Australian National University, Canberra, 1995Google Scholar
  28. 28.
    D. Taunt, Remarks on the isomorphism problem in theories of construction of finite groups. Proc. Camb. Philos. Soc. 51, 16–24 (1955)MathSciNetCrossRefGoogle Scholar
  29. 29.
    The GAP Group, GAP – groups, algorithms and programming, Version 4.4. Available from (2005)
  30. 30.
    M. Vaughan-Lee, B. Eick, SglPPow – Database of certain p-groups (2016). A GAP 4 package, see [29]Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.TU BraunschweigBraunschweigGermany
  2. 2.Justus-Liebig-Universität GießenGießenGermany
  3. 3.Colorado State UniversityFort CollinsUSA

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