q=5 (q mod 8 = 5): 8 soluble classes of S, 7 with PGL-invariant class; witnesses for (p=l, delta): ['4[1^1,2^3]:b2', '12[1^1,2^3,3^8]:b2']  [0.0s]
    X=3[1^1,3^2]                               b1=0 b2=0  |N_S|=6 |F|=3, beta trivial on N/F: 6 of 6
    X=5[1^1,5^4]                               b1=0 b2=0  O_l(N) != 1
    X=2[1^1,2^1]                               b1=0 b2=0  |N_S|=4 |F|=4, beta trivial on N/F: 4 of 4
    X=4[1^1,2^3]                               b1=0 b2=1  |N_S|=12 |F|=4, beta trivial on N/F: 0 of 12
    X=6[1^1,2^3,3^2]                           b1=0 b2=0  |N_S|=6 |F|=3, beta trivial on N/F: 6 of 6
    X=12[1^1,2^3,3^8]                          b1=0 b2=1  |N_S|=12 |F|=4, beta trivial on N/F: 0 of 12
    X=10[1^1,2^5,5^4]                          b1=0 b2=0  O_l(N) != 1
q=7 (q mod 8 = 7): 14 soluble classes of S, 7 with PGL-invariant class; witnesses for (p=l, delta): NONE  [0.0s]
    X=7[1^1,7^6]                               b1=0 b2=0  O_l(N) != 1
    X=3[1^1,3^2]                               b1=0 b2=0  |N_S|=6 |F|=3, beta trivial on N/F: 6 of 6
    X=2[1^1,2^1]                               b1=0 b2=0  |N_S|=8 |F|=8, beta trivial on N/F: 8 of 8
    X=4[1^1,2^1,4^2]                           b1=0 b2=0  |N_S|=8 |F|=8, beta trivial on N/F: 8 of 8
    X=6[1^1,2^3,3^2]                           b1=0 b2=0  |N_S|=6 |F|=3, beta trivial on N/F: 6 of 6
    X=8[1^1,2^5,4^2]                           b1=0 b2=0  |N_S|=8 |F|=8, beta trivial on N/F: 8 of 8
    X=21[1^1,3^14,7^6]                         b1=0 b2=0  O_l(N) != 1
q=9 (q mod 8 = 1): 19 soluble classes of S, 8 with PGL-invariant class; witnesses for (p=l, delta): NONE  [0.0s]
    X=5[1^1,5^4]                               b1=0 b2=0  |N_S|=10 |F|=5, beta trivial on N/F: 10 of 10
    X=2[1^1,2^1]                               b1=0 b2=0  |N_S|=8 |F|=8, beta trivial on N/F: 8 of 8
    X=4[1^1,2^1,4^2]                           b1=0 b2=0  |N_S|=8 |F|=8, beta trivial on N/F: 8 of 8
    X=9[1^1,3^8]                               b1=0 b2=0  O_l(N) != 1
    X=8[1^1,2^5,4^2]                           b1=0 b2=0  |N_S|=8 |F|=8, beta trivial on N/F: 8 of 8
    X=10[1^1,2^5,5^4]                          b1=0 b2=0  |N_S|=10 |F|=5, beta trivial on N/F: 10 of 10
    X=18[1^1,2^9,3^8]                          b1=0 b2=0  O_l(N) != 1
    X=36[1^1,2^9,3^8,4^18]                     b1=0 b2=0  O_l(N) != 1
q=11 (q mod 8 = 3): 13 soluble classes of S, 10 with PGL-invariant class; witnesses for (p=l, delta): ['4[1^1,2^3]:b2', '12[1^1,2^3,3^8]:b2']  [0.0s]
    X=3[1^1,3^2]                               b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=11[1^1,11^10]                            b1=0 b2=0  O_l(N) != 1
    X=5[1^1,5^4]                               b1=0 b2=0  |N_S|=10 |F|=5, beta trivial on N/F: 10 of 10
    X=2[1^1,2^1]                               b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=4[1^1,2^3]                               b1=0 b2=1  |N_S|=12 |F|=4, beta trivial on N/F: 0 of 12
    X=6[1^1,2^1,3^2,6^2]                       b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=12[1^1,2^3,3^8]                          b1=0 b2=1  |N_S|=12 |F|=4, beta trivial on N/F: 0 of 12
    X=10[1^1,2^5,5^4]                          b1=0 b2=0  |N_S|=10 |F|=5, beta trivial on N/F: 10 of 10
    X=12[1^1,2^7,3^2,6^2]                      b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=55[1^1,5^44,11^10]                       b1=0 b2=0  O_l(N) != 1
q=13 (q mod 8 = 5): 15 soluble classes of S, 12 with PGL-invariant class; witnesses for (p=l, delta): ['4[1^1,2^3]:b2', '12[1^1,2^3,3^8]:b2']  [0.1s]
    X=3[1^1,3^2]                               b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=7[1^1,7^6]                               b1=0 b2=0  |N_S|=14 |F|=7, beta trivial on N/F: 14 of 14
    X=2[1^1,2^1]                               b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=13[1^1,13^12]                            b1=0 b2=0  O_l(N) != 1
    X=6[1^1,2^1,3^2,6^2]                       b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=4[1^1,2^3]                               b1=0 b2=1  |N_S|=12 |F|=4, beta trivial on N/F: 0 of 12
    X=12[1^1,2^3,3^8]                          b1=0 b2=1  |N_S|=12 |F|=4, beta trivial on N/F: 0 of 12
    X=12[1^1,2^7,3^2,6^2]                      b1=0 b2=0  |N_S|=12 |F|=6, beta trivial on N/F: 12 of 12
    X=14[1^1,2^7,7^6]                          b1=0 b2=0  |N_S|=14 |F|=7, beta trivial on N/F: 14 of 14
    X=39[1^1,3^26,13^12]                       b1=0 b2=0  O_l(N) != 1
    X=26[1^1,2^13,13^12]                       b1=0 b2=0  O_l(N) != 1
    X=78[1^1,2^13,3^26,6^26,13^12]             b1=0 b2=0  O_l(N) != 1
nice -n 10 python3 -u -I  5 7 9 11 13  0.14s user 0.01s system 98% cpu 0.146 total
