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For every countable ordinal α, we construct an $l_1$-predual $X_{\alpha }$ which is isometric to a subspace of $C\left({\omega }^{{\omega }^{{\omega }^{\alpha }+2}}\right)$ and isomorphic to a quotient of $C\left({\omega }^{\omega }\right)$. However, $X_{\alpha }$ is not isomorphic to a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$.