Stiefel–Whitney classes for finite symplectic groups

Let [Formula: see text] be an odd prime power, and [Formula: see text] the finite symplectic group. We give an expression for the total Stiefel–Whitney Classes (SWCs) for orthogonal representations [Formula: see text] of [Formula: see text], in terms of character values of [Formula: see text] at elements of order [Formula: see text]. We give “universal formulas” for the fourth and eighth SWCs. For [Formula: see text], we compute the subring of the mod [Formula: see text] cohomology generated by the SWCs [Formula: see text].