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The Solvable Primitive Permutation Groups of Degree at Most 6560

Published online by Cambridge University Press:  01 February 2010

B. Eick  and
B. Höfling
B. Eick
Affiliation:
Institut für Geometrie, Technische Universität, Pockelsstr. 14, 38106 Braunschweig, Germany [email protected], http://www.tu-bs.de/~beick
B. Höfling
Affiliation:
Institut für Geometrie, Technische Universität, Pockelsstr. 14, 38106 Braunschweig, Germany [email protected], http://www.tu-bs.de/~bhoeflin

Abstract

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The authors present an algorithm to construct conjugacy class representatives of the solvable primitive subgroups of Sd for a given degree d. Using this method, they determine the solvable primitive permutation groups of degree at most 6560 (that is, 38 – 1), up to conjugacy.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 29 - 39
Copyright
Copyright © London Mathematical Society 2003

References

1Aschbacher, Michael, ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984) 469–514.CrossRefGoogle Scholar
2Dixon, John D. and Mortimer, Brian, ‘The primitive permutation groups of degree less than 1000’, Math. Proc. Cambridge Philos. Soc. 103 (1988) 213–238.CrossRefGoogle Scholar
3Doerk, Klaus and Hawkes, Trevor, Finite solvable groups (De Gruyter, Berlin/NewYork, 1992).CrossRefGoogle Scholar
4Eick, Bettina, Leedham-Green, C. R. and O'Brien, E. A., ‘Constructing automorphism groups of p-groups’, Comm. Algebra 30 (2002) 2271–2295.Google Scholar
5Felsch, Volkmar and Neubüser, Joachim, ‘On a programme for the determination of the automorphism group of a finite group’, Computational Problems in Abstract Algebra (Proc.Conf., Oxford, 1967) (Pergamon, Oxford, 1970) 59–60.Google Scholar
6THE GAP GROUP, GAP- Groups, Algorithms, and Programming, Version 4.3 (The GAP Group, Aachen⁄St Andrews, 2002); http://www.gap-system.org.Google Scholar
7Holt, D. F. and Rees, Sarah, ‘Testing for isomorphism between finitely presented groups’, Groups, combinatorics & geometry (Durham, 1990), London Math. Soc. Lecture Note Ser. 165 (Cambridge Univ. Press, Cambridge, 1992) 459–75.Google Scholar
8Holt, Derek F. and Rees, Sarah, ‘Testing modules for irreducibility’, J. Austral. Math. Soc. Ser. A 57 (1994) 1–16.Google Scholar
9Holt, Derek F., Leedham-Green, Charles R., O'Brien, Eamonn A. and Rees, Sarah, ‘Computing matrix group decompositions with respect to a normal subgroup’, J. Algebra 184 (1996) 818–838.Google Scholar
10Hulpke, Alexander, ‘Konstruktion transitiver Permutationsgruppen’, Dissertation, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1996.Google Scholar
11Hulpke, Alexander, ‘Computing subgroups invariant under a set of automorphisms’, J. Symb. Comput. 27 (1999) 415–27.CrossRefGoogle Scholar
12Ivanyos, Gábor and Lux, Klaus, ‘Treating the exceptional cases of the MeatAxe’, Experiment.Math. 9 (2000) 373–381.CrossRefGoogle Scholar
13Laue, Reinhard, Neubüser, Joachim and Schoenwaelder, Ulrich, ‘Algorithms for finite soluble groups and the SOGOS system’, Computational Group Theory,Proceedings LMS Symposium on Computational Group Theory, Durham 1982 (ed. Atkinson, Michael D., Academic Press, London, 1984) 105–135.Google Scholar
14Liebeck, Martin W., Praeger, Cheryl E. and Saxl, Jan, ‘On the O'Nan-Scott theorem for finite primitive permutation groups’, J. Austral. Math. Soc. Ser. A 44 (1988) 389–396.Google Scholar
15Mecky, Matthias and Neubüser, Joachim, ‘Some remarks on the computation of conjugacy classes of soluble groups’, Bull. Austral. Math. Soc. 40 (1989) 281–292.CrossRefGoogle Scholar
16O'Brien, E. A., ‘Isomorphism testing for p-groups’, J. Symbolic Comput. 17 (1) (1994) 133–147.Google Scholar
17Roney-Dougal, Colva M. and Unger, William R., ‘The affine primitive permutation groups of degree less than 1000’, J. Symbolic Comput., to appear.Google Scholar
18Short, Mark W., The primitive soluble permutation groups of degree less than 256, Lecture Notes in Math. 1519 (Springer, Berlin/Heidelberg, 1992).CrossRefGoogle Scholar
19Sims, Charles C., ‘Computing the order of a solvable permutation group’, J. Symbolic Comput. 9 (1990) 699–705.Google Scholar
20Sims, Charles C., Computation with finitely presented groups (Cambridge Univ.Press, Cambridge, 1994).Google Scholar
21Smith, Michael J., ‘Computing automorphisms of finite soluble groups’, PhD thesis, Australian National University, Canberra, Australia, 1994.Google Scholar
22Suprunenko, Dimitriĭ A., Matrix groups, Transl. Math. Monogr. 45 (Amer. Math. Soc., Providence, RI, 1976).CrossRefGoogle Scholar
23Taylor, Donald E., The geometry of the classical groups (Heldermann Verlag, Berlin, 1992).Google Scholar
24Theissen, Heiko, ‘Eine Methode zur Normalisatorberechnung in Permutationsgruppen mit Anwendungen in der Konstruktion primitiver Gruppen’, Dissertation, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1997.Google Scholar