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There are source and target morphisms $\\phi\\colon \\H_{g,G,\\xi}\\to \\M_{g,r}$ and $\\delta\\colon \\H_{g,G,\\xi}\\to \\M_{g',b}$, remembering the curves $C$ and $D$ together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $\\phi_* [\\H_{g,G,\\xi}]$. 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