A000019 Number of primitive permutation groups of degree n.

1, 1, 2, 2, 5, 4, 7, 7, 11, 9, 8, 6, 9, 4, 6, 22, 10, 4, 8, 4, 9, 4, 7, 5, 28, 7, 15, 14, 8, 4, 12, 7, 4, 2, 6, 22, 11, 4, 2, 8, 10, 4, 10, 4, 9, 2, 6, 4, 40, 9, 2, 3, 8, 4, 8, 9, 5, 2, 6, 9, 14, 4, 8, 74, 13, 7, 10, 7, 2, 2, 10, 4, 16, 4, 2, 2, 4, 6, 10, 4, 155, 10, 6, 6, 6, 2, 2, 2, 10, 4, 10, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• core
• nice

Comments

A check found errors in Theißen's data (degree 121 and 125) as well as in Short's work (degree 169). - Alexander Hulpke, Feb 19 2002

There is an error at n=574 in the Dixon-Mortimer paper. - Colva M. Roney-Dougal.

References

• CRC Handbook of Combinatorial Designs, 1996, pp. 595ff.
• K. Harada and H. Yamaki, The irreducible subgroups of GL_n(2) with n <= 6, C. R. Math. Rep. Acad. Sci. Canada 1, 1979, 75-78.
• A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
• M. W. Short, The Primitive Soluble Permutation Groups of Degree less than 256, LNM 1519, 1992, Springer
• 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).
• H. Theißen, Eine Methode zur Normalisatorberechnung in Permutationsgruppen mit Anwendungen in der Konstruktion primitiver Gruppen, Dissertation, RWTH, RWTH-A, 1997 [But see comment above about errors! ]

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..4095, computed using the Magma command shown below (terms 1..2499 from N. J. A. Sloane, computed using the GAP command shown below, which uses the results of Colva M. Roney-Dougal, a(1575) corrected).
• Soleyman Askary, Nader Biranvand, and Farrokh Shirjian, New constructions of orbit codes based on imprimitive wreath products and wreathed tensor products, Rend. Circ. Mat. Palermo Ser. II (2023).
• J. D. Dixon and B. Mortimer, The primitive permutation groups of degree less than 1000, Math. Proc. Cambridge Philos. Soc., 103, 213-238, 1988 [But see comment above about errors! ]
• 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]
• A. Hulpke, Transitive groups of small degree
• A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
• 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.
• C. C. Sims, Letter to N. J. A. Sloane (no date)
• Index entries for sequences related to groups
• Index entries for "core" sequences

Crossrefs

Cf. A000001, A023675, A023676, A000638, A002106, A005432, A000637.

Programs

GAP

List([2..2499],NrPrimitiveGroups);

Magma

[NumberOfPrimitiveGroups(i) : i in [1..4095]];

Extensions

More terms and additional references from Alexander Hulpke