The non-split extension Ḡ = 25· GL4(2) is a subgroup of the Dempwolff group 25· GL5(2) of index 496 and has order 645120. In this paper, the author computes the character table of Ḡ using the technique of Fischer-Clifford matrices. Very interesting results are obtained on the projective character tables of the inertia factor groups of Ḡ.