GAP Manual: 51.15. Congruences

Congruences( tbl, chars, prime_powermap, prime )
Congruences( tbl, chars, prime_powermap, prime, "quick" )

improves the parametrized map prime_powermap (see More about Maps and Parametrized Maps) that is an approximation of the prime-th powermap of the character table tbl:

For G a group with character table tbl, g inG and a character chi of tbl, the congruence [GaloisCyc( chi(g), prime ) equivchi(g^prime)pmodprime] holds; if the representative order of g is relative prime to prime, we have [GaloisCyc( chi(g), prime ) = chi(g^prime).] Congruences checks these congruences for the (virtual ) characters in the list chars.

If "quick" is specified, only those classes are considered for which prime_powermap is ambiguous.

If there are classes for which no image is possible, false is returned, otherwise Congruences returns true.

    # see example in "InitPowermap"
    gap> Congruences( t, t.irreducibles, pow, 2 ); pow;
    true
    [ 1, 1, 3, 4, 5, [ 2, 16 ], [ 2, 16, 17 ], 8, 3, 4, 11, 12,
      [ 6, 7, 18, 19 ], 14, [ 9, 20 ], 1, 1, 2, 2, 3, 4, 5,
      [ 6, 7, 18, 19 ], 8, 9, 9, [ 10, 21 ], 11, 12, 16, 16, [ 2, 16 ],
      [ 2, 16 ], 17, 17, [ 6, 18 ], [ 6, 18 ], [ 6, 7, 18, 19 ],
      [ 6, 7, 18, 19 ], 20, 20, [ 9, 20 ], [ 9, 20 ], 22, 22, 24, 24,
      [ 15, 25, 26 ], [ 15, 25, 26 ], 28, 28, 29, 29 ]
    # continued in "ConsiderKernels"

Congruences is used by Powermap Powermap.