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 $H_{d}(G_{1}\times \cdots \times G_{k}, \chi_{1}\times\cdots \times \chi_{k})$ and investigate the existence of an o-basis of this type of classes.

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