Abstract
This chapter discusses the characters of the Weyl group E8. The group F of order 192.10! = 21435527 = 696,729,600 whose 112 absolutely irreducible characters (all rational) are described in this chapter is isomorphic to the Weyl group E8. The group F itself is described by Coxeter as the eight-dimensional group 3[4,2,1] of symmetries of Gosset's semi-regular polytope 421, and it is the largest of the irreducible finite groups generated by reflections. Its factor group A = F/C with respect to its center C = {I, −I} is the orthogonal group of half the order investigated by Hamill as a collineation group and by Edge as the group A of automorphisms of the nonsingular quadric consisting of 135 points of a finite projective space. The simple group denoted FH(8, 2) by Dickson is a subgroup A+ of index 2 in A = F/C.
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