A129520 - OEIS

%I #5 May 01 2013 21:08:56

%S 3,6,24,40,60,120

%N Orders of symmetric groups associated with the exceptional groups G2, F4, E6, E7, E7.5, E8, E9 (or better? E11): A_3, S_3, A_4, S_5, "? ", A_5, S_5.

%C The calculation by hand based on dimensions of: Dim_Ex(i)-n(i)*Dim_Symm(i)=small_integer The groups E7.5 and E9 are better thought of as E7.6 and E11 when integer results are wanted. The "?" question mark group is a symmetrical group between s4 and A5 with dimension 40; it is not in any of the books I have or on the web as far as I can find by searching.

%D Gareth Jones and David Singerman, Bull. London Math Soc. 28, (1996) pages 561-590 ( A_4, S_4, A_5 on p585)

%e 14-4*3=2

%e 52-8*6=4

%e 78-5*12=6

%e 133-5*24=13

%e 190-4.6*40=6

%e 248-4*60=8

%e 484-4*120=4

%e Example:

%e DimEx(E7.6)/Dim_cartan=190/7.6=25

%e DimEx(E11)/Dim_cartan=484/11=44

%e which fit in the sequence of the traditional exceptional groups A118889

%e as:3, 7, 13, 13, 19, 25, 31,44

%Y Cf. A118889.

%K nonn,uned,obsc

%O 1,1

%A _Roger L. Bagula_, May 28 2007

%E Partially edited by _N. J. A. Sloane_, May 15 2010