Galois number fields with small root discriminant

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω≈44.7632. We definitively treat the cases G=A4, A5, A6 and S4, S5, S6, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois group SL3(2), A7, S7, PGL2(7), SL2(8), ΣL2(8), PGL2(9), PΓL2(9), PSL2(11), and , and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K(G,Ω) is empty.