symmetry classes of polynomials associated with the ‎direct ‎product of permutation groups

‎let $g_{i} $ be a subgroup of $ s_{m_{i}}‎ ,‎ 1 leq i leq k$‎. ‎suppose $chi_{i}$ is an irreducible complex character of $g_{i}$‎. ‎we consider $ g_{1}times cdots times g_{k} $ as subgroup of $ s_{m} $‎, ‎where $ m=m_{1}+cdots‎ +‎m_{k} $‎. ‎in this paper‎, ‎we give a formula for the dimension of $h_{d}(g_{1}times cdots times g_{k}‎, ‎chi_{1}timescdots times chi_{k})$ and investigate the existence of an o-basis of this type of classes‎.