The orthogonal simple group 0  (3) has three conjugacy classes of maximal subgroups of the form 36:L4(3). These groups are all isomorphic to each other and each group has order 4421589120 with index 1120 in 0 (3). In this paper, we will compute the ordinary carácter table of one of these classes of maximal subgroups using the technique of Fischer-Clifford matrices. This technique is very efficient to compute the ordinary character table of an extension group Ḡ = N.G and especially where the normal subgroup N of Ḡ is an elementary abelian p-group. The said technique reduces the computation of the ordinary character table of Ḡ to find a handful of so-called Fischer-Clifford matrices of Ḡ and the ordinary or projective character tables of the inertia factor groups of the action of Ḡ on N.