How Fast Can We Compute Orbits of Groups?

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)


Many problems in Combinatorics and related fields reduce to the problem of computing orbits of groups acting on finite sets. One of the techniques is known under the name Snakes and Ladders. We offer the alternate name poset classification algorithm. We will describe this technique and compare the performance on example problems.


Group orbits Classification Combinatorial object 


  1. 1.
    Al-Azemi, A., Betten, A., Chowdhury, S.R.: A rainbow clique search algorithm for BLT-sets. In: Davenport, J.H., Kauers, M., Labahn, G., Urban, J. (eds.) ICMS 2018. LNCS, vol. 10931, pp. 71–79. Springer, Cham (2018)Google Scholar
  2. 2.
    Bamberg, J., Betten, A., Cara, Ph., De Beule, J., Lavrauw, M., Neunhöffer, M.: Finite Incidence Geometry. FinInG - a GAP package, version 1.4 (2017)Google Scholar
  3. 3.
    Betten, A.: Classifying discrete objects with orbiter. ACM Commun. Comput. Algebra 47(3/4), 183–186 (2014). Scholar
  4. 4.
    Betten, A.: Orbiter - a program to classify discrete objects (2016–2018).
  5. 5.
    Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-Correcting Linear Codes, Classification by Isometry and Applications. Algorithms and Computation in Mathematics, vol. 18. Springer, Heidelberg (2006). Scholar
  6. 6.
    Betten, A.: Rainbow cliques and the classification of small BLT-sets. In: Kauers, M. (ed.) ISSAC 2013, 26–29 June 2013, Boston, Massachusetts, pp. 53–60 (2013)Google Scholar
  7. 7.
    Betten, A., Hirschfeld, J.W.P., Karaoglu, F.: Classification of cubic surfaces with twenty-seven lines over the finite field of order thirteen. Eur. J. Math. 4(1), 37–50 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Betten, A., Karaoglu, F.: Cubic surfaces over small finite fields. Submitted to Designs, Codes and CryptographyGoogle Scholar
  9. 9.
    Betten, A.: Classifying cubic surfaces over finite fields using orbiter. In: Davenport, J.H., Kauers, M., Labahn, G., Urban, J. (eds.) ICMS 2018. LNCS, vol. 10931, pp. 55–61. Springer, Cham (2018)Google Scholar
  10. 10.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Braun, M.: Some new designs over finite fields. Bayreuth. Math. Schr. 74, 58–68 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.10 (2018).
  13. 13.
    Kaski, P., Östergård, P.: Classification Algorithms for Codes and Designs. Algorithms and Computation in Mathematics, vol. 15. Springer, Heidelberg (2006). Scholar
  14. 14.
    Koch, M.: Neue Strategien zur Lösung von Isomorphieproblemen. (German) [New strategies for the solution of isomorphism problems] Ph.D. thesis. University of Bayreuth (2015)Google Scholar
  15. 15.
    Leon, J.S.: Partitions, refinements, and permutation group computation. In: Groups and Computation, II (New Brunswick, NJ, 1995), vol. 28. DIMACS Series Discrete Mathematics Theoretical Computer Science, pp. 123–158. American Mathematical Society, Providence (1997)Google Scholar
  16. 16.
    McKay, B.D.: Isomorph-free exhaustive generation. J. Algorithms 26(2), 306–324 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    McKay, B.D., Piperno, A.: Practical graph isomorphism II. J. Symbolic Comput. 60, 94–112 (2014). Scholar
  18. 18.
    Schmalz, B.: Verwendung von Untergruppenleitern zur Bestimmung von Doppelnebenklassen. (German) [Use of subgroup ladders for the determination of double cosets]. Bayreuth. Math. Schr. 31, 109–143 (1990)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Colorado State UniversityFort CollinsUSA

Personalised recommendations