The following table lists, for each Galois group (that is, each transitive permutation group, up to conjugacy in the symmetric group of the same degree) up to degree 9, an irreducible polynomial having the given group as its Galois group over the rationals.
| Group | Order | Polynomial |
|---|---|---|
| 2T1 = S(2) | 2 | X^2-X+1 |
| 3T1 = A(3) | 3 | X^3-X^2-2*X+1 |
| 3T2 = S(3) | 6 | X^3-X^2+1 |
| 4T1 = C(4) | 4 | X^4-X^3+X^2-X+1 |
| 4T2 = E(4) | 4 | X^4-X^2+1 |
| 4T3 = D(4) | 8 | X^4-2*X^3+X-1 |
| 4T4 = A(4) | 12 | X^4-2*X^3+2*X^2+2 |
| 4T5 = S(4) | 24 | X^4-X^3+1 |
| 5T1 = C(5) | 5 | X^5-X^4-4*X^3+3*X^2+3*X-1 |
| 5T2 = D(5) | 10 | X^5-2*X^4+2*X^3-X^2+1 |
| 5T3 = F(5) | 20 | X^5-9*X^3-4*X^2+17*X+12 |
| 5T4 = A(5) | 60 | X^5-X^4+2*X^2-2*X+2 |
| 5T5 = S(5) | 120 | X^5-X^3-2*X^2+1 |
| 6T1 = C(6) | 6 | X^6-X^5+X^4-X^3+X^2-X+1 |
| 6T2 = D_6(6) | 6 | X^6-3*X^5-2*X^4+9*X^3-5*X+1 |
| 6T3 = S(3)[x]2 | 12 | X^6+X^4-2*X^3+X^2-X+1 |
| 6T4 = A_4(6) | 12 | X^6+X^4-2*X^2-1 |
| 6T5 = F_18(6) | 18 | X^6-3*X^5+4*X^4-2*X^3+X^2-X+1 |
| 6T6 = 2A_4(6) | 24 | X^6-2*X^5+2*X^3-X-1 |
| 6T7 = [2^2]S(3) | 24 | X^6+X^4-1 |
| 6T8 = 1/2[2^3]S(3) | 24 | X^6+4*X^5-9*X^4-51*X^3-46*X^2+8 |
| 6T9 = F_18(6):2 | 36 | X^6-3*X^5+4*X^4-X^3+X^2-2*X+7 |
| 6T10 = 1/2[S(3)^2]2 | 36 | X^6-X^5+X^4-X^3-4*X^2+5 |
| 6T11 = [2^3]S(3) | 48 | X^6-X^5-X^3-X+1 |
| 6T12 = PSL(2,5) | 60 | X^6-10*X^4-7*X^3+15*X^2+14*X+3 |
| 6T13 = F_36(6):2 | 72 | X^6-2*X^5+2*X^4-X+1 |
| 6T14 = PGL(2,5) | 120 | X^6+3*X^4-2*X^3+6*X^2+1 |
| 6T15 = A(6) | 360 | X^6-2*X^4+X^2-2*X-1 |
| 6T16 = S(6) | 720 | X^6-X^5+X^3-X^2+1 |
| 7T1 = C(7) | 7 | X^7-X^6-12*X^5+7*X^4+28*X^3-14*X^2-9*X-1 |
| 7T2 = D(7) | 14 | X^7-2*X^6+2*X^5+X^3-3*X^2+X-1 |
| 7T3 = F_21(7) | 21 | X^7-8*X^5-2*X^4+16*X^3+6*X^2-6*X-2 |
| 7T4 = F_42(7) | 42 | X^7-3*X^6+9*X^5-13*X^4+17*X^3-10*X^2+4*X+1 |
| 7T5 = L(7) | 168 | X^7-8*X^5-2*X^4+15*X^3+4*X^2-6*X-2 |
| 7T6 = A(7) | 2520 | X^7-2*X^6+4*X^4-5*X^3+2*X-1 |
| 7T7 = S(7) | 5040 | X^7-X^6-X^5+X^3+X^2-X-1 |
| 8T1 = C(8) | 8 | X^8+8*X^6+20*X^4+16*X^2+2 |
| 8T2 = 4[x]2 | 8 | X^8+2*X^6+4*X^4+8*X^2+16 |
| 8T3 = E(8) | 8 | X^8-X^4+1 |
| 8T4 = D_8(8) | 8 | X^8-4*X^6+37*X^4-66*X^2+64 |
| 8T5 = Q_8(8) | 8 | X^8+12*X^6+36*X^4+36*X^2+9 |
| 8T6 = D(8) | 16 | X^8-3*X^5-X^4+3*X^3+1 |
| 8T7 = 1/2[2^3]4 | 16 | X^8+10*X^6+25*X^4+20*X^2+5 |
| 8T8 = 2D_8(8) | 16 | X^8+24*X^6+126*X^4+216*X^2+117 |
| 8T9 = E(8):2 | 16 | X^8+2*X^4-3*X^2+1 |
| 8T10 = [2^2]4 | 16 | X^8-13*X^6+44*X^4-17*X^2+1 |
| 8T11 = 1/2[2^3]E(4) | 16 | X^8-X^5-2*X^4+4*X^2+X+1 |
| 8T12 = 2A_4(8) | 24 | X^8-2*X^7+X^6+X^5-X^4+2*X^3+4*X^2-16*X+16 |
| 8T13 = E(8):3 | 24 | X^8-3*X^6+3*X^4+2*X^2+1 |
| 8T14 = 1/2(S_4[x]2) | 24 | X^8-26*X^6+99*X^4-126*X^2+49 |
| 8T15 = [1/4cD(4)^2]2 | 32 | X^8-X^4-1 |
| 8T16 = 1/2[2^4]4 | 32 | X^8-2*X^7-2*X^6+X^5+5*X^4+X^3+3*X^2+3*X+1 |
| 8T17 = [4^2]2 | 32 | X^8+12*X^6+48*X^4+68*X^2+17 |
| 8T18 = E(8):E_4 | 32 | X^8+3*X^6+3*X^4+3*X^2+1 |
| 8T19 = [1/4eD(4)^2]2 | 32 | X^8+X^6+2*X^2+4 |
| 8T20 = [2^3]4 | 32 | X^8-4*X^6-6*X^4+4*X^2+1 |
| 8T21 = [1/4dD(4)^2]2 | 32 | X^8+2*X^4-4*X^2+2 |
| 8T22 = E(8):D_4 | 32 | X^8+13*X^6+47*X^4+44*X^2+4 |
| 8T23 = 2S_4(8) | 48 | X^8+X^7-3*X^6+X^5+8*X^4+X^2+7*X+1 |
| 8T24 = E(8):D_6 | 48 | X^8-X^7+X^6+X^2+X+1 |
| 8T25 = F_56(8) | 56 | X^8-4*X^7+8*X^6-6*X^5+2*X^4+6*X^3-3*X^2+X+3 |
| 8T26 = 1/2[2^4]eD(4) | 64 | X^8-5*X^5-3*X^4-5*X^3+1 |
| 8T27 = [2^4]4 | 64 | X^8-2*X^7+3*X^5-X^4-3*X^3+2*X+1 |
| 8T28 = 1/2[2^4]dD(4) | 64 | X^8-2*X^6+3*X^4+2 |
| 8T29 = E(8):D_8 | 64 | X^8-X^6-X^4+X^2+1 |
| 8T30 = 1/2[2^4]cD(4) | 64 | X^8-4*X^6-20*X^4+14 |
| 8T31 = [2^4]E(4) | 64 | X^8+12*X^6+48*X^4+72*X^2+31 |
| 8T32 = [2^3]A(4) | 96 | X^8-2*X^6-13*X^4-9*X^2+4 |
| 8T33 = [1/3A(4)^2]2 | 96 | X^8-2*X^7-4*X^5+12*X^4+2*X^3-14*X^2-5*X+11 |
| 8T34 = 1/2[E(4)^2:S_3]2 | 96 | X^8+6*X^6-4*X^5-31*X^4-12*X^3+96*X^2-26*X+29 |
| 8T35 = [2^4]D(4) | 128 | X^8+2*X^6-12*X^4-3*X^2+11 |
| 8T36 = E(8):F_21 | 168 | X^8+3*X^7+20*X^4+18*X^3-18*X^2-8*X+14 |
| 8T37 = L(8) | 168 | X^8-4*X^7+7*X^6-7*X^5+7*X^4-7*X^3+7*X^2+5*X+1 |
| 8T38 = [2^4]A(4) | 192 | X^8-2*X^6-7*X^4+3*X^2+8 |
| 8T39 = [2^3]S(4) | 192 | X^8+X^4+X^2+1 |
| 8T40 = 1/2[2^4]S(4) | 192 | X^8-X^7+4*X^5-2*X^4+3*X^2-X+1 |
| 8T41 = [E(4)^2:S_3]2 | 192 | X^8-2*X^6-4*X^5+4*X^3-10*X^2-8*X-1 |
| 8T42 = [A(4)^2]2 | 288 | X^8-2*X^7+2*X^6-2*X^5+2*X^4-X+1 |
| 8T43 = L(8):2 | 336 | X^8-X^6-3*X^5-X^4+4*X^3+4*X^2-2*X-1 |
| 8T44 = [2^4]S(4) | 384 | X^8-X^5-X^4-X^3+1 |
| 8T45 = [1/2S(4)^2]2 | 576 | X^8-2*X^6+7*X^4-8*X^2-4*X+7 |
| 8T46 = 1/2[S(4)^2]2 | 576 | X^8-X^7+X^5-4*X^4+5*X^3+6*X^2-2*X-1 |
| 8T47 = [S(4)^2]2 | 1152 | X^8-2*X^7+2*X^6+X^3+X+1 |
| 8T48 = AL(8) | 1344 | X^8+3*X^7-X^6-10*X^5-9*X^4-X^3+7*X^2+11*X+4 |
| 8T49 = A(8) | 20160 | X^8-2*X^7+3*X^5-5*X^4+2*X^3+2*X^2-X+1 |
| 8T50 = S(8) | 40320 | X^8-X^6-X^5-3*X^4+4*X^2-1 |
| 9T1 = C(9) | 9 | X^9-X^8-8*X^7+7*X^6+21*X^5-15*X^4-20*X^3+10*X^2+5*X-1 |
| 9T2 = 3[x]3 | 9 | X^9-15*X^7-4*X^6+54*X^5+12*X^4-38*X^3-9*X^2+6*X+1 |
| 9T3 = D(9) | 18 | X^9-3*X^8+4*X^7-5*X^6+6*X^5-X^4-5*X^3+4*X^2-2 |
| 9T4 = S(3)[x]3 | 18 | X^9-5*X^8-X^7+4*X^6+2*X^5+3*X^4-X^3-3*X^2+1 |
| 9T5 = S(3)[1/2]S(3) | 18 | X^9-3*X^6+3*X^3+1 |
| 9T6 = 1/3[3^3]3 | 27 | X^9-3*X^8-10*X^7+42*X^6-28*X^5-28*X^4+28*X^3+2*X^2-6*X+1 |
| 9T7 = [3^2]3 | 27 | X^9-3*X^8-21*X^7+78*X^5+69*X^4-21*X^3-39*X^2-12*X-1 |
| 9T8 = S(3)[x]S(3) | 36 | X^9-X^8+3*X^6+X^5+X^4+3*X^3+2*X^2+1 |
| 9T9 = E(9):4 | 36 | X^9-45*X^7-93*X^6+72*X^5+216*X^4+63*X^3-81*X^2-54*X-9 |
| 9T10 = [3^2]S(3)_6 | 54 | X^9+6*X^8+15*X^7+18*X^6+11*X^5+X^4-2*X^3+3*X^2-X+1 |
| 9T11 = 1/2[3^2:2]S(3) | 54 | X^9-3*X^6+3*X^3+8 |
| 9T12 = [3^2]S(3) | 54 | X^9-4*X^8+4*X^7+4*X^6-7*X^5-2*X^4+4*X^3+3*X^2-X-1 |
| 9T13 = [1/2S(3)^2]3 | 54 | X^9-X^6-2*X^3+1 |
| 9T14 = E(9):Q_8 | 72 | X^9-3*X^8+12*X^7-12*X^6+12*X^5-12*X^4+12*X^3-12*X^2+9*X-3 |
| 9T15 = E(9):8 | 72 | X^9-9*X^7-21*X^6+72*X^5+99*X^4-99*X^3-585*X^2+549*X+166 |
| 9T16 = E(9):D_8 | 72 | X^9-X^8-2*X^6-X^5+3*X^4+X^2+X-1 |
| 9T17 = 3 wr 3 | 81 | X^9-4*X^8-2*X^7+22*X^6-14*X^5-22*X^4+20*X^3+2*X^2-5*X+1 |
| 9T18 = [1/2S(3)^2]S(3) | 108 | X^9-X^3-1 |
| 9T19 = E(9):2D_8 | 144 | X^9-3*X^8+6*X^7-18*X^6+12*X^5-24*X^4+24*X^3-12*X^2+6*X-2 |
| 9T20 = [3^3]S(3) | 162 | X^9-4*X^8+6*X^7-8*X^6+7*X^5-4*X^4+2*X^3+1 |
| 9T21 = 1/2[3^3:2]S(3) | 162 | X^9-4*X^8+6*X^7-9*X^6+13*X^5+3*X^4-24*X^3+15*X^2-5*X+5 |
| 9T22 = [3^3:2]3 | 162 | X^9-4*X^8+8*X^7-11*X^6+9*X^5-3*X^4+5*X^3-4*X^2-X+1 |
| 9T23 = E(9):2A_4 | 216 | X^9-3*X^8+X^6+15*X^5-13*X^4-3*X^3+4*X-1 |
| 9T24 = [3^3:2]S(3) | 324 | X^9+X^8+3*X^7+3*X^5-5*X^4+6*X^3+X+1 |
| 9T25 = [1/2S(3)^3]3 | 324 | X^9-3*X^8+3*X^7+4*X^6-12*X^5+9*X^4+X^3-9*X^2+6*X-1 |
| 9T26 = E(9):2S_4 | 432 | X^9-X^7-5*X^6+X^5+2*X^4+4*X^3-3*X^2-X+1 |
| 9T27 = PSL(2,8) | 504 | X^9+X^7-4*X^6-12*X^4-X^3-7*X^2-X-1 |
| 9T28 = [S(3)^3]3 | 648 | X^9-2*X^8+3*X^7-X^6-2*X^5+5*X^4-4*X^3+2*X-1 |
| 9T29 = [1/2S(3)^3]S(3) | 648 | X^9-3*X^6-5*X^5+5*X^2-1 |
| 9T30 = 1/2[S(3)^3]S(3) | 648 | X^9-X^8+2*X^5-2*X^4+2*X^2-2*X+1 |
| 9T31 = [S(3)^3]S(3) | 1296 | X^9-5*X^7-X^6+7*X^5+4*X^4-2*X^3-5*X^2-X+1 |
| 9T32 = P|L(2,8) | 1512 | X^9-X^8-4*X^7+28*X^3+26*X^2+9*X+1 |
| 9T33 = A(9) | 181440 | X^9-X^8-X^7-2*X^5+4*X^4-5*X^2+1 |
| 9T34 = S(9) | 362880 | X^9-4*X^7+X^5-2*X^4+5*X^3+3*X^2-2*X-1 |