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%I M1316 N0504 #90 Dec 13 2020 18:44:09
%S 1,1,2,5,5,16,7,50,34,45,8,301,9,63,104,1954,10,983,8,1117,164,59,7,
%T 25000,211,96,2392,1854,8,5712,12,2801324,162,115,407,121279,11,76,
%U 306,315842,10,9491,10,2113,10923,56,6
%N Number of transitive permutation groups of degree n.
%C It is conjectured that this is the number of Galois groups for irreducible polynomials of order n. (All such Galois groups are transitive.) - _Charles R Greathouse IV_, May 28 2014
%D G. Butler and J. McKay, personal communication.
%D C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. Butler and J. McKay, <a href="http://dx.doi.org/10.1080/00927878308822884">The transitive groups of degree up to eleven</a>, Comm. Algebra, 11 (1983), 863-911.
%H G. Butler and J. McKay, <a href="/A000637/a000637_1.pdf">The transitive groups of degree up to eleven</a>, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]
%H John J. Cannon and Derek F. Hol, <a href="http://www.warwick.ac.uk/~mareg/download/papers/trans32/trans32.pdf">The transitive permutation groups of degree 32</a>
%H F. N. Cole, <a href="http://dx.doi.org/10.1090/S0002-9904-1893-00137-7">Note on the substitution groups of six, seven, and eight letters</a>, Bull. Amer. Math. Soc. 2 (1893), 184-190. Gives a(8)=48 instead of 50.
%H Computational Algebra Group, <a href="https://magma.maths.usyd.edu.au/magma/releasenotes/2/21/#subsection_10_8">Summary of New Features in Magma V2.21</a>
%H J. Conway, A. Hulpke, and J. McKay, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.792">On Transitive Permutation Groups</a>, LMS Journal of Computation and Mathematics 1 (1998), pp. 1-8. See especially Appendix A.
%H D. Holt, <a href="/A000019/a000019_1.pdf">Enumerating subgroups of the symmetric group</a>, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]
%H Derek Holt and Gordon Royle, <a href="https://arxiv.org/abs/1811.09015">A Census of Small Transitive Groups and Vertex-Transitive Graphs</a>, arXiv:1811.09015 [math.CO], 2018.
%H A. Hulpke, <a href="http://www.math.colostate.edu/~hulpke/smalldeg.html">Transitive groups of small degree</a>
%H A. Hulpke, <a href="http://www.math.colostate.edu/~hulpke/paper/prom.ps.gz">Konstruktion transitiver Permutationsgruppen</a>, Dissertation, RWTH Aachen, 1996.
%H A. Hulpke, <a href="http://dx.doi.org/10.1016/j.jsc.2004.08.002">Constructing transitive permutation groups</a>, J. Symbolic Comput. 39 (2005), 1-30.
%H E. G. Köhler, F. H. Lutz, <a href="http://arXiv.org/abs/math.CO/0506520">Triangulated manifolds with few vertices: Vertex-transitive triangulations</a>, arXiv:math/0506520 [math.GT], 2005.
%H J. Labelle and Y. N. Yeh, <a href="http://dx.doi.org/10.1016/0097-3165(89)90019-8">The relation between Burnside rings and combinatorial species</a>, J. Combin. Theory, A 50 (1989), 269-284. See page 280.
%H G. A. Miller, <a href="http://dx.doi.org/10.1090/S0002-9904-1896-00327-X ">On the lists of all the substitution groups that can be formed with a given number of elements</a>, Bull. Amer. Math. Soc., 2 (1896), 138-145.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Inverse_Galois_problem">Inverse Galois problem</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%e a(3)=2: A_3 and S_3.
%o (GAP) a:=function(n)
%o return Length(AllTransitiveGroups(NrMovedPoints,n));
%o end; # _Charles R Greathouse IV_, May 28 2014
%Y Cf. A000001, A000019, A177244, A186277.
%K nonn,core,hard,more,nice
%O 1,3
%A _N. J. A. Sloane_
%E Corrected and extended to degree 31 by _Alexander Hulpke_, Aug 15 1996
%E Further corrections from _Alexander Hulpke_, Feb 19 2002
%E Degree 32 extended by _Artur Jasinski_, Feb 17 2011
%E Extended to degree 47 by _Gabriel Verret_, May 07 2016
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