GAP Manual: 54.4. BergerCondition

BergerCondition( chi )
BergerCondition( G )

Called with an irreducible character chi of the group G of degree d, BergerCondition returns true if chi satisfies M^(prime) leqker(chi) for every normal subgroup M of G with the property that M leqker(psi) for all psiinIrr(G) with psi(1) < chi(1), and false otherwise.

Called with a group G, BergerCondition returns true if all irreducible characters of G satisfy the inequality above, and false otherwise; in the latter case InfoMonomial tells about the smallest degree for that the inequality is violated.

For groups of odd order the answer is always true by a theorem of T. R. Berger (see Ber76, Thm. 2.2).

    gap> BergerCondition( S4 );
    true
    gap> BergerCondition( Sl23 );
    false
    gap> List( Irr( Sl23 ), BergerCondition );
    [ true, true, true, false, false, false, true ]
    gap> List( Irr( Sl23 ), Degree );
    [ 1, 1, 1, 2, 2, 2, 3 ]