Geodesic
Determining the Frattini Subgroup from the Character Table
Canadian journal of mathematics, Tome 28 (1976) no. 3, pp. 560-567

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Brauer [1, p. 141] has discussed the question of which subgroups of a group can be determined from its character table. He mentions, referring to p-groups, that the Frattini subgroup can be determined. We show that for an arbitrary finite solvable group, the Frattini subgroup can be determined from the character table. Then we exhibit an infinite set of pairs of non-solvable groups such that both members of a given pair have the same character table but Frattini subgroups of different orders. All groups to be considered are finite. 
Garrison, Sidney. Determining the Frattini Subgroup from the Character Table. Canadian journal of mathematics, Tome 28 (1976) no. 3, pp. 560-567. doi: 10.4153/CJM-1976-055-0
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