Coxeter covers of the classical coxeter groups

Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either B_n or D_n. Let C_Y(T) be a natural quotient of C(T), and if C(T) is simply-laced (which means all the relations between the generators has order 2 or 3), C _Y(T) is a generalized Coxeter group, too. Let A_t,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that C_Y(T) is isomorphic to A_t,n ⋊ B_n or A_t,n ⋊ D_n, depends on whether the signed graph T contains loops or not, or in other words C(T) is simply-laced or not, and t is the number of the cycles in T. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.