CharacterTable for symmetric groups $S_n$ with large $n$

Expanding a comment: we can use GAP to compute large character tables and then import them to Mathematica.

A simple GAP function based on this solution:

storeSymmetricCharacterTable := function(n, filename)
    local c, f;
    c := CharacterTable("Symmetric", n);
    f := OutputTextFile(filename, false);
    SetPrintFormattingStatus(f, false);
    PrintTo(f, "SymmetricCharacterTable[", n, "] =n");
    PrintTo(f, "  {"CharacterParameters"->");
    PrintTo(f, "{", JoinStringsWithSeparator(List(CharacterParameters(c), x->Concatenation("{",JoinStringsWithSeparator(x[2]),"}"))), "},n");
    PrintTo(f, "   "CharacterTable"->");
    PrintTo(f, "{", JoinStringsWithSeparator(List(Irr(c), x->Concatenation("{",JoinStringsWithSeparator(x),"}"))), "}}n");
    CloseStream(f);
end;;

Store this function in a file ssct.gap, then start GAP and run

Read("ssct.gap");
storeSymmetricCharacterTable(3, "symm_3.txt");

which now contains Mathematica-format data:

SymmetricCharacterTable[3] =
  {"CharacterParameters"->{{1,1,1},{2,1},{3}},
   "CharacterTable"->{{1,-1,1},{2,0,-1},{1,1,1}}}

Compare to Mathematica code:

FiniteGroupData[{"SymmetricGroup", 3}, "ConjugacyClassNames"]
(*    {"{1,1,1}", "{2,1}", "{3}"}    *)

FiniteGroupData[{"SymmetricGroup", 3}, "CharacterTable"]
(*    {{1, 1, 1}, {2, 0, -1}, {1, -1, 1}}    *)

Notice that the signs in the character tables differ between Mathematica and GAP.