%N Orders of simple groups which are non-cyclic and non-alternating.
%C From _Bernard Schott_, Apr 26 2020: (Start)
%C About a(16) = 20160; 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but, 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8.
%C Indeed, 20160 is the smallest order for which there exist two nonisomorphic simple groups and it is the order of this group PSL_3(4) that was missing in the data. The first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1900) [see the link]. (End)
%D L. E. Dickson, Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
%H David Madore, <a href="http://www.madore.org/~david/math/simplegroups.html#table1">Orders of non abelian simple groups</a>
%H Ida May Schottenfels, <a href="https://www.jstor.org/stable/1967281">Two non isomorphic simple groups of the same order 20160</a>, Annals of Mathematics, Second Series, Vol. 1, No. 1/4 (1900), pp. 147-152.
%e From _Bernard Schott_, Apr 27 2020: (Start)
%e Two particular examples:
%e a(1) = 168 is the order of the smallest non-cyclic and non-alternating simple group, this Lie group is the projective special linear group PSL_2(7) that is isomorphic to the general linear group GL_3(2).
%e a(12) = 7920 is the order of the smallest sporadic group (A001228), the Mathieu group M_11. (End)
%Y Cf. A001034, A001710, A005180, A109379.
%Y Subsequence: A001228 (sporadic groups).
%A _Artur Jasinski_, Feb 16 2008
%E More terms from _R. J. Mathar_, Apr 23 2009
%E a(16) = 20160 inserted by _Bernard Schott_, Apr 26 2020
%E Incorrect formula and programs removed by _R. J. Mathar_, Apr 27 2020
%E Terms checked by _Bernard Schott_, Apr 26 2020