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A002106 Number of transitive permutation groups of degree n.
(Formerly M1316 N0504)
1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301, 9, 63, 104, 1954, 10, 983, 8, 1117, 164, 59, 7, 25000, 211, 96, 2392, 1854, 8, 5712, 12, 2801324, 162, 115, 407, 121279, 11, 76, 306, 315842, 10, 9491, 10, 2113, 10923, 56, 6 (list; graph; refs; listen; history; text; internal format)



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


G. Butler and J. McKay, personal communication.

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=1..47.

G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911.

G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]

John J. Cannon and Derek F. Hol, The transitive permutation groups of degree 32

F. N. Cole, Note on the substitution groups of six, seven, and eight letters, Bull. Amer. Math. Soc. 2 (1893), 184-190. Gives a(8)=48 instead of 50.

Computational Algebra Group, Summary of New Features in Magma V2.21

J. Conway, A. Hulpke, and J. McKay, On Transitive Permutation Groups, LMS Journal of Computation and Mathematics 1 (1998), pp. 1-8. See especially Appendix A.

D. Holt, Enumerating subgroups of the symmetric group, 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]

Derek Holt and Gordon Royle, A Census of Small Transitive Groups and Vertex-Transitive Graphs, arXiv:1811.09015 [math.CO], 2018.

A. Hulpke, Transitive groups of small degree

A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.

A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.

E. G. Köhler, F. H. Lutz, Triangulated manifolds with few vertices: Vertex-transitive triangulations, arXiv:math/0506520 [math.GT], 2005.

J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284. See page 280.

G. A. Miller, On the lists of all the substitution groups that can be formed with a given number of elements, Bull. Amer. Math. Soc., 2 (1896), 138-145.

Wikipedia, Inverse Galois problem

Index entries for sequences related to groups

Index entries for "core" sequences


a(3)=2: A_3 and S_3.


(GAP) a:=function(n)

return Length(AllTransitiveGroups(NrMovedPoints, n));

end; # Charles R Greathouse IV, May 28 2014


Cf. A000001, A000019, A177244, A186277.

Sequence in context: A154698 A063786 A121304 * A232316 A184604 A064630

Adjacent sequences:  A002103 A002104 A002105 * A002107 A002108 A002109




N. J. A. Sloane


Corrected and extended to degree 31 by Alexander Hulpke, Aug 15 1996

Further corrections from Alexander Hulpke, Feb 19 2002

Degree 32 extended by Artur Jasinski, Feb 17 2011

Extended to degree 47 by Gabriel Verret, May 07 2016



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Last modified March 1 06:03 EST 2021. Contains 341732 sequences. (Running on oeis4.)