For almost simple groups, the character table determines if Sylow 3-subgroups are 2-generated, shown equivalent to a Galois action condition on principal 3-block characters.
Malle , Die unipotenten Charaktere von \,^2 F _4(q^2) , Comm
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Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups
For almost simple groups, the character table determines if Sylow 3-subgroups are 2-generated, shown equivalent to a Galois action condition on principal 3-block characters.