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If F is a group, then the extensions G of F of order o with G/phi(G)=F, where phi(G) is the Frattini subgroup, are called Frattini extensions.
A group given by G/phi(G), where phi(G) is the Frattini subgroup of a given group G.
The intersection phi(G) of all maximal subgroups of a given group G.

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