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Is there any group with the non-trivial frattini subgroup such that the intersection of the Frattini subgroup and the commutator subgroup is trivial? Expect the groups can be constructed by the direct product of two groups; an abelian group with the non-trivial Frattini subgroup and a group with the trivial Fratini subgroup?

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Take $G=S \times C_{p^2}$, where $S$ is a finite non-abelian simple group and $p$ is prime. It is well-known that the Frattini subgroup of a direct product is the direct product of the Frattini subgroups of the composing factors: $\Phi(G)=\{1\} \times C_p$. Also, $G'=S \times \{1\}$. So $\Phi(G) \cap G'=1$ as wanted.

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