Characters of factor representations of finite type of the wreath products $G = \mathfrak{S}_{\infty}(T)$ of any compact groups $T$ with the infinite symmetric group $\mathfrak{S}_{\infty}$ were explicitly given in [HH4]-[HH6], as the extremal continuous positive definite class functions $f_{A}$ on $G$ determined by a parameter $A$. In this paper, we give a special kind of realization of a factor representation $\pi ^{A}$ associated to $f_{A}$. This realization is better than the Gelfand-Raikov realization $\pi _{f}$, $f = f_{A}$, in [GR] at least at the point where a matrix element $\langle \pi ^{A}(g)v_{0}, v_{0}\rangle$ of $\pi ^{A}$ for a cyclic vector $v_{0}$ can be calculated explicitly, which is exactly equal to the character $f_{A}$ (and so $\pi ^{A}$ has a trace-element $v_{0}$). So the positive-definiteness of class functions $f_{A}$ given in [HH4]-[HH6] is automatically guaranteed, a proof of which occupies the first half of [HH6] in the case of $T$ infinite. The case where $T$ is abelian contains the cases of infinite Weyl groups and the limits $\mathfrak{S}_{\infty}(\mathbf{Z}_{r}) = \lim _{n\to\infty}G(r,1,n)$ of complex reflexion groups.