The character table of a sharply $5$-transitive subgroup of the alternating group of degree 12

Document Type : Research Paper


Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U. K.


We calculate the character table of a sharply $5$-transitive subgroup of Alt(12)‎, ‎and of a sharply $4$-transitive subgroup of Alt(11)‎. ‎Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups‎, ‎and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations‎.


Main Subjects

[1] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups. Maximal subgroups
and ordinary characters for simple groups. With comput. assist. from J. G. Thackray., Oxford: Clarendon Press.,
[2] G. Frobenius, Uber die Charaktere der mehrfach transitiven Gruppen, ¨ Berl. Ber., 1904 (1904) 558–571.
[3] G. James and M. Liebeck, Representations and characters of groups. 2nd ed, 2nd ed. ed., Cambridge: Cambridge
University Press, 2000.
[4] F. Ladisch, What did Frobenius prove about M12? (answer), MathOverflow.
[5] K. Lux and H. Pahlings, Representations of groups. A computational approach, 124, Cambridge: Cambridge University Press, 2010.
[6] E. Mathieu, Sur la fonction cinq fois transitive de 24 quantit´es, Liouville J. (2), 18 (1873) 25–47.
[7] G. A. Miller, On the supposed five-fold transitive function of 24 elements and 19! ÷ 48 values, Messenger (2) 27
(1897) 187–190.
[8] , Sur plusieurs groupes simples., Bull. Soc. Math. Fr., 28 (1900) 266–267.
[9] J. Saxl, The complex characters of the symmetric groups that remain irreducible in subgroups, J. Algebra, 111
(1987) 210–219.
[10] G. Frobenius and I. Schur, Uber die reellen Darstellungen der endlichen Gruppen, Berl. Ber. (1906), 186–208.