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Let $G=C_{19}\rtimes C_9=\langle a,b\ |\ a^{19}=b^9=1, a^b=a^7\rangle$ be a nonabelian group of order $171$. Is there a (compact) 3-manifold $M$ with $\pi_1(M)\cong G$?

Thanks for any help!

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    @Jesse: I am listing all possible finite groups that occur as $\pi_1$ of a compact 3-manifold. It is groups of these type I am having the hardest time eliminating (without using any of the theory of Seifert-fibred spaces).2011-08-30

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If $\pi_1(M) = G$, then the universal cover of $M$ is a compact (because $G$ is finite) simply connected $3$-manifold, hence (by Poincare) a $3$-sphere, on which $G$ acts. The spherical space form conjecture then implies that $G$ is a group of isometeries of $S^3$, i.e. that $G$ is a subgroup of $O(4)$. (See also this wikipedia entry.) Since $G$ has odd order, in fact $G$ is a subgroup of $SO(4)$, or of $PSO(4)$. I don't think that your group $G$ embeds into $PSO(4)$ (use the isomorphism $PSO(4) \cong PSO(3)\times PSO(3)$ and the known list of finite subgroups of $PSO(3)$), and so I think the answer is no.

(I don't know enough about the area to know if this kind of question can be answered without appealing to the full machinery of Poincare/geometrization.)

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    I would also add that you need to be careful when applying the Poincare conjecture: you need $M$ closed!2011-08-30