Special Properties of the Linear Group GL(3, q)

A square matrix over f with non-negative integral trace is called a quasipermutationmatrix over f. let g be a finite linear group of degree n, that is, afinite group of authomorphisms of an n-dimensional complex vector space or,equivalently, a finite group of non-singular matrices of order n with complexcoefficients. we shall say that g is a quasi-permutation group if the trace of everyelement of g is a non-negative rational integer. thus a permutation group of degreen has a representation as a quasi-permutation group of degree n.ln this paper wecalculate the kernel and galois groups for the irreducible characters of the groupgl(3, q). these help us to obtain the permutation and quasi-permutationrepresentations of the group gl(3, q) .