The group G considered in
this paper is the finite irreducible unitary reflection group in 6 dimensions denoted by
[321]3 in the notation of H. S. M. Coxeter. A character table for G is constructed.
There are 169 irreducible characters and it is shown that each character has Schur
index 1 over Q. Furthermore, each character has values in the field Q() and this
field is a splitting field for G.