q=  4 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=120, |C(U)|=4]
q=  4 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q=  4 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q=  4 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=6]
q=  4 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=10]
q=  4 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=3]
q=  4 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=5]
q=  4 PASS Lemma 3.1 r=2 k=1: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 10, |C| = 6]
q=  4 PASS coset sigma^1 (r=2) p=2: Lemma 3.4 (Q_5, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=5, |c|=60 elements tested]
q=  4 PASS coset sigma^1 (r=2) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q=  4 PASS coset sigma^1 (r=2) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q=  4 PASS coset sigma^1 (r=2) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q=  4 done in 0.0 s
q=  5 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=120, |C(U)|=5]
q=  5 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q=  5 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q=  5 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=6]
q=  5 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=10]
q=  5 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=3]
q=  5 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q=  5 PASS (P3) q=+-3 mod 8: Sylow 2 = Klein, one class, |N_S(V)|=12 (A4), |N_PGL(V)|=24  [1 class(es), |N_S(V)|=12, |N_PGL(V)|=24]
q=  5 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=5, |c|=60 elements tested]
q=  5 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=5, |c|=60 elements tested]
q=  5 PASS coset delta (r=2) p=5: Lemma 3.5(b) (V, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q=  5 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=5, |c|=60 elements tested]
q=  5 done in 0.0 s
q=  7 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=336, |C(U)|=7]
q=  7 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q=  7 PASS (Cl') N_S(Sylow 2) soluble  [|P|=8, |N|=8]
q=  7 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=6]
q=  7 PASS (Cl') N_S(Sylow 7) soluble  [|P|=7, |N|=21]
q=  7 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=3]
q=  7 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q=  7 PASS (P3) q=+-1 mod 8: two classes of Klein four-groups swapped by delta, |N_S(V)|=24, Sylow 2 dihedral of order >= 8, self-normalizing  [2 classes, |N_S(V)|=24, |P|=8, |N_S(P)|=8]
q=  7 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=168 elements tested]
q=  7 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=168 elements tested]
q=  7 PASS Remark 3.6: excluded pair (p=l=7, c=delta): (b1) fails for all Sylow t, t odd != l
q=  7 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=168 elements tested]
q=  7 done in 0.0 s
q=  8 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=1512, |C(U)|=8]
q=  8 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q=  8 PASS (Cl') N_S(Sylow 2) soluble  [|P|=8, |N|=56]
q=  8 PASS (Cl') N_S(Sylow 3) soluble  [|P|=9, |N|=18]
q=  8 PASS (Cl') N_S(Sylow 7) soluble  [|P|=7, |N|=14]
q=  8 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=9]
q=  8 PASS (P2) t=7: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=7]
q=  8 PASS Lemma 3.1 r=3 k=1: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 84, |C| = 6]
q=  8 PASS Lemma 3.1 r=3 k=2: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 84, |C| = 6]
q=  8 PASS coset sigma^1 (r=3) p=2: Lemma 3.4 (Q_7, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=504 elements tested]
q=  8 PASS coset sigma^1 (r=3) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=56, |F*(D)|=8, 56 elements of c normalize X]
q=  8 PASS coset sigma^1 (r=3) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=56, |F*(D)|=8, 56 elements of c normalize X]
q=  8 PASS coset sigma^1 (r=3) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=56, |F*(D)|=8, 56 elements of c normalize X]
q=  8 PASS coset sigma^2 (r=3) p=2: Lemma 3.4 (Q_7, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=504 elements tested]
q=  8 PASS coset sigma^2 (r=3) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=56, |F*(D)|=8, 56 elements of c normalize X]
q=  8 PASS coset sigma^2 (r=3) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=56, |F*(D)|=8, 56 elements of c normalize X]
q=  8 PASS coset sigma^2 (r=3) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=56, |F*(D)|=8, 56 elements of c normalize X]
q=  8 done in 0.1 s
q=  9 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=1440, |C(U)|=9]
q=  9 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q=  9 PASS (Cl') N_S(Sylow 2) soluble  [|P|=8, |N|=8]
q=  9 PASS (Cl') N_S(Sylow 3) soluble  [|P|=9, |N|=36]
q=  9 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=10]
q=  9 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=5]
q=  9 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_5
q=  9 PASS (P3) q=+-1 mod 8: two classes of Klein four-groups swapped by delta, |N_S(V)|=24, Sylow 2 dihedral of order >= 8, self-normalizing  [2 classes, |N_S(V)|=24, |P|=8, |N_S(P)|=8]
q=  9 PASS Lemma 3.1 r=2 k=1: C_PGL(phi0^k)=PGL(2,3); all elements of order r of the coset conjugate  [#order r = 30, |C| = 24]
q=  9 PASS Lemma 3.1: no involution in (PGL - PSL)phi0 (coset delta sigma^(f/2))
q=  9 PASS coset sigma^1 (r=2) p=2: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=36, |F*(D)|=9, 36 elements of c normalize X]
q=  9 PASS coset sigma^1 (r=2) p=3: Lemma 3.4 (Q_5, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=5, |c|=360 elements tested]
q=  9 PASS coset sigma^1 (r=2) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=36, |F*(D)|=9, 36 elements of c normalize X]
q=  9 PASS coset sigma^1 (r=2) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=36, |F*(D)|=9, 36 elements of c normalize X]
q=  9 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=9, |c|=360 elements tested]
q=  9 PASS Remark 3.6: excluded pair (p=l=3, c=delta): (b1) fails for all Sylow t, t odd != l
q=  9 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=9, |c|=360 elements tested]
q=  9 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=9, |c|=360 elements tested]
q=  9 PASS coset deltasigma^1 (r=2) p=2: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=36, |F*(D)|=9, 36 elements of c normalize X]
q=  9 PASS coset deltasigma^1 (r=2) p=3: Lemma 3.4 (Q_5, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=5, |c|=360 elements tested]
q=  9 PASS coset deltasigma^1 (r=2) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=36, |F*(D)|=9, 36 elements of c normalize X]
q=  9 PASS coset deltasigma^1 (r=2) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=36, |F*(D)|=9, 36 elements of c normalize X]
q=  9 done in 0.1 s
q= 11 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=1320, |C(U)|=11]
q= 11 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 11 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q= 11 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=12]
q= 11 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=10]
q= 11 PASS (Cl') N_S(Sylow 11) soluble  [|P|=11, |N|=55]
q= 11 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=6]
q= 11 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 11 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=5]
q= 11 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_5
q= 11 PASS (P3) q=+-3 mod 8: Sylow 2 = Klein, one class, |N_S(V)|=12 (A4), |N_PGL(V)|=24  [1 class(es), |N_S(V)|=12, |N_PGL(V)|=24]
q= 11 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=660 elements tested]
q= 11 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=660 elements tested]
q= 11 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=660 elements tested]
q= 11 PASS coset delta (r=2) p=11: Lemma 3.5(b) (V, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q= 11 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=660 elements tested]
q= 11 done in 0.0 s
q= 13 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=2184, |C(U)|=13]
q= 13 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 13 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q= 13 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=12]
q= 13 PASS (Cl') N_S(Sylow 7) soluble  [|P|=7, |N|=14]
q= 13 PASS (Cl') N_S(Sylow 13) soluble  [|P|=13, |N|=78]
q= 13 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=6]
q= 13 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 13 PASS (P2) t=7: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=7]
q= 13 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_7
q= 13 PASS (P3) q=+-3 mod 8: Sylow 2 = Klein, one class, |N_S(V)|=12 (A4), |N_PGL(V)|=24  [1 class(es), |N_S(V)|=12, |N_PGL(V)|=24]
q= 13 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=13, |X|=13, |c|=1092 elements tested]
q= 13 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=13, |X|=13, |c|=1092 elements tested]
q= 13 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=13, |X|=13, |c|=1092 elements tested]
q= 13 PASS coset delta (r=2) p=13: Lemma 3.5(b) (V, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q= 13 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=13, |X|=13, |c|=1092 elements tested]
q= 13 done in 0.1 s
q= 16 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=16320, |C(U)|=16]
q= 16 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 16 PASS (Cl') N_S(Sylow 2) soluble  [|P|=16, |N|=240]
q= 16 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=30]
q= 16 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=30]
q= 16 PASS (Cl') N_S(Sylow 17) soluble  [|P|=17, |N|=34]
q= 16 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=15]
q= 16 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=15]
q= 16 PASS (P2) t=17: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=17]
q= 16 PASS Lemma 3.1 r=2 k=1: C_PGL(phi0^k)=PGL(2,4); all elements of order r of the coset conjugate  [#order r = 68, |C| = 60]
q= 16 PASS coset sigma^2 (r=2) p=2: Lemma 3.4 (Q_17, b1) witness per Definition 2.9, (Cl')  [t=17, |X|=17, |c|=4080 elements tested]
q= 16 PASS coset sigma^2 (r=2) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=240, |F*(D)|=16, 240 elements of c normalize X]
q= 16 PASS coset sigma^2 (r=2) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=240, |F*(D)|=16, 240 elements of c normalize X]
q= 16 PASS coset sigma^2 (r=2) p=17: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=240, |F*(D)|=16, 240 elements of c normalize X]
q= 16 PASS coset sigma^2 (r=2) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=240, |F*(D)|=16, 240 elements of c normalize X]
q= 16 done in 0.5 s
q= 17 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=4896, |C(U)|=17]
q= 17 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 17 PASS (Cl') N_S(Sylow 2) soluble  [|P|=16, |N|=16]
q= 17 PASS (Cl') N_S(Sylow 3) soluble  [|P|=9, |N|=18]
q= 17 PASS (Cl') N_S(Sylow 17) soluble  [|P|=17, |N|=136]
q= 17 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=9]
q= 17 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 17 PASS (P3) q=+-1 mod 8: two classes of Klein four-groups swapped by delta, |N_S(V)|=24, Sylow 2 dihedral of order >= 8, self-normalizing  [2 classes, |N_S(V)|=24, |P|=16, |N_S(P)|=16]
q= 17 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=17, |X|=17, |c|=2448 elements tested]
q= 17 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=17, |X|=17, |c|=2448 elements tested]
q= 17 PASS Remark 3.6: excluded pair (p=l=17, c=delta): (b1) fails for all Sylow t, t odd != l
q= 17 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=17, |X|=17, |c|=2448 elements tested]
q= 17 done in 0.2 s
q= 19 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=6840, |C(U)|=19]
q= 19 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 19 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q= 19 PASS (Cl') N_S(Sylow 3) soluble  [|P|=9, |N|=18]
q= 19 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=20]
q= 19 PASS (Cl') N_S(Sylow 19) soluble  [|P|=19, |N|=171]
q= 19 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=9]
q= 19 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 19 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=10]
q= 19 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_5
q= 19 PASS (P3) q=+-3 mod 8: Sylow 2 = Klein, one class, |N_S(V)|=12 (A4), |N_PGL(V)|=24  [1 class(es), |N_S(V)|=12, |N_PGL(V)|=24]
q= 19 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=19, |X|=19, |c|=3420 elements tested]
q= 19 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=19, |X|=19, |c|=3420 elements tested]
q= 19 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=19, |X|=19, |c|=3420 elements tested]
q= 19 PASS coset delta (r=2) p=19: Lemma 3.5(b) (V, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q= 19 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=19, |X|=19, |c|=3420 elements tested]
q= 19 done in 0.4 s
q= 23 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=12144, |C(U)|=23]
q= 23 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 23 PASS (Cl') N_S(Sylow 2) soluble  [|P|=8, |N|=8]
q= 23 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=24]
q= 23 PASS (Cl') N_S(Sylow 11) soluble  [|P|=11, |N|=22]
q= 23 PASS (Cl') N_S(Sylow 23) soluble  [|P|=23, |N|=253]
q= 23 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=12]
q= 23 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 23 PASS (P2) t=11: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=11]
q= 23 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_11
q= 23 PASS (P3) q=+-1 mod 8: two classes of Klein four-groups swapped by delta, |N_S(V)|=24, Sylow 2 dihedral of order >= 8, self-normalizing  [2 classes, |N_S(V)|=24, |P|=8, |N_S(P)|=8]
q= 23 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=23, |X|=23, |c|=6072 elements tested]
q= 23 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=23, |X|=23, |c|=6072 elements tested]
q= 23 PASS coset delta (r=2) p=11: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=23, |X|=23, |c|=6072 elements tested]
q= 23 PASS Remark 3.6: excluded pair (p=l=23, c=delta): (b1) fails for all Sylow t, t odd != l
q= 23 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=23, |X|=23, |c|=6072 elements tested]
q= 23 done in 0.8 s
q= 25 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=31200, |C(U)|=25]
q= 25 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 25 PASS (Cl') N_S(Sylow 2) soluble  [|P|=8, |N|=8]
q= 25 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=24]
q= 25 PASS (Cl') N_S(Sylow 5) soluble  [|P|=25, |N|=300]
q= 25 PASS (Cl') N_S(Sylow 13) soluble  [|P|=13, |N|=26]
q= 25 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=12]
q= 25 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 25 PASS (P2) t=13: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=13]
q= 25 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_13
q= 25 PASS (P3) q=+-1 mod 8: two classes of Klein four-groups swapped by delta, |N_S(V)|=24, Sylow 2 dihedral of order >= 8, self-normalizing  [2 classes, |N_S(V)|=24, |P|=8, |N_S(P)|=8]
q= 25 PASS Lemma 3.1 r=2 k=1: C_PGL(phi0^k)=PGL(2,5); all elements of order r of the coset conjugate  [#order r = 130, |C| = 120]
q= 25 PASS Lemma 3.1: no involution in (PGL - PSL)phi0 (coset delta sigma^(f/2))
q= 25 PASS coset sigma^1 (r=2) p=2: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset sigma^1 (r=2) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset sigma^1 (r=2) p=5: Lemma 3.4 (Q_13, b1) witness per Definition 2.9, (Cl')  [t=13, |X|=13, |c|=7800 elements tested]
q= 25 PASS coset sigma^1 (r=2) p=13: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset sigma^1 (r=2) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=25, |c|=7800 elements tested]
q= 25 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=25, |c|=7800 elements tested]
q= 25 PASS Remark 3.6: excluded pair (p=l=5, c=delta): (b1) fails for all Sylow t, t odd != l
q= 25 PASS coset delta (r=2) p=13: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=25, |c|=7800 elements tested]
q= 25 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=5, |X|=25, |c|=7800 elements tested]
q= 25 PASS coset deltasigma^1 (r=2) p=2: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset deltasigma^1 (r=2) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset deltasigma^1 (r=2) p=5: Lemma 3.4 (Q_13, b1) witness per Definition 2.9, (Cl')  [t=13, |X|=13, |c|=7800 elements tested]
q= 25 PASS coset deltasigma^1 (r=2) p=13: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 PASS coset deltasigma^1 (r=2) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=300, |F*(D)|=25, 300 elements of c normalize X]
q= 25 done in 2.3 s
q= 27 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=58968, |C(U)|=27]
q= 27 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 27 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q= 27 PASS (Cl') N_S(Sylow 3) soluble  [|P|=27, |N|=351]
q= 27 PASS (Cl') N_S(Sylow 7) soluble  [|P|=7, |N|=28]
q= 27 PASS (Cl') N_S(Sylow 13) soluble  [|P|=13, |N|=26]
q= 27 PASS (P2) t=7: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=14]
q= 27 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_7
q= 27 PASS (P2) t=13: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=13]
q= 27 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_13
q= 27 PASS (P3) q=+-3 mod 8: Sylow 2 = Klein, one class, |N_S(V)|=12 (A4), |N_PGL(V)|=24  [1 class(es), |N_S(V)|=12, |N_PGL(V)|=24]
q= 27 PASS Lemma 3.1 r=3 k=1: C_PGL(phi0^k)=PGL(2,3); all elements of order r of the coset conjugate  [#order r = 819, |C| = 24]
q= 27 PASS Lemma 3.1 r=3 k=2: C_PGL(phi0^k)=PGL(2,3); all elements of order r of the coset conjugate  [#order r = 819, |C| = 24]
q= 27 PASS coset sigma^1 (r=3) p=2: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^1 (r=3) p=3: Lemma 3.4 (Q_7, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=9828 elements tested]
q= 27 PASS coset sigma^1 (r=3) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^1 (r=3) p=13: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^1 (r=3) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^2 (r=3) p=2: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^2 (r=3) p=3: Lemma 3.4 (Q_7, b1) witness per Definition 2.9, (Cl')  [t=7, |X|=7, |c|=9828 elements tested]
q= 27 PASS coset sigma^2 (r=3) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^2 (r=3) p=13: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset sigma^2 (r=3) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=351, |F*(D)|=27, 351 elements of c normalize X]
q= 27 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=27, |c|=9828 elements tested]
q= 27 PASS coset delta (r=2) p=3: Lemma 3.5(b) (V, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q= 27 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=27, |c|=9828 elements tested]
q= 27 PASS coset delta (r=2) p=13: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=27, |c|=9828 elements tested]
q= 27 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=3, |X|=27, |c|=9828 elements tested]
q= 27 done in 9.6 s
q= 29 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=24360, |C(U)|=29]
q= 29 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 29 PASS (Cl') N_S(Sylow 2) soluble  [|P|=4, |N|=12]
q= 29 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=30]
q= 29 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=30]
q= 29 PASS (Cl') N_S(Sylow 7) soluble  [|P|=7, |N|=28]
q= 29 PASS (Cl') N_S(Sylow 29) soluble  [|P|=29, |N|=406]
q= 29 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=15]
q= 29 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 29 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=15]
q= 29 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_5
q= 29 PASS (P2) t=7: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=14]
q= 29 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_7
q= 29 PASS (P3) q=+-3 mod 8: Sylow 2 = Klein, one class, |N_S(V)|=12 (A4), |N_PGL(V)|=24  [1 class(es), |N_S(V)|=12, |N_PGL(V)|=24]
q= 29 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=29, |X|=29, |c|=12180 elements tested]
q= 29 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=29, |X|=29, |c|=12180 elements tested]
q= 29 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=29, |X|=29, |c|=12180 elements tested]
q= 29 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=29, |X|=29, |c|=12180 elements tested]
q= 29 PASS coset delta (r=2) p=29: Lemma 3.5(b) (V, b2) witness per Definition 2.9, (Cl')  [|D|=12, |F*(D)|=4, 12 elements of c normalize X]
q= 29 PASS coset delta (r=2) p=11: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=29, |X|=29, |c|=12180 elements tested]
q= 29 done in 1.7 s
q= 31 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=29760, |C(U)|=31]
q= 31 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 31 PASS (Cl') N_S(Sylow 2) soluble  [|P|=32, |N|=32]
q= 31 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=30]
q= 31 PASS (Cl') N_S(Sylow 5) soluble  [|P|=5, |N|=30]
q= 31 PASS (Cl') N_S(Sylow 31) soluble  [|P|=31, |N|=465]
q= 31 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=15]
q= 31 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_3
q= 31 PASS (P2) t=5: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=15]
q= 31 PASS Remark 3.6 (b1 part): some element of delta centralizes Q_5
q= 31 PASS (P3) q=+-1 mod 8: two classes of Klein four-groups swapped by delta, |N_S(V)|=24, Sylow 2 dihedral of order >= 8, self-normalizing  [2 classes, |N_S(V)|=24, |P|=32, |N_S(P)|=32]
q= 31 PASS coset delta (r=2) p=2: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=31, |X|=31, |c|=14880 elements tested]
q= 31 PASS coset delta (r=2) p=3: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=31, |X|=31, |c|=14880 elements tested]
q= 31 PASS coset delta (r=2) p=5: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=31, |X|=31, |c|=14880 elements tested]
q= 31 PASS Remark 3.6: excluded pair (p=l=31, c=delta): (b1) fails for all Sylow t, t odd != l
q= 31 PASS coset delta (r=2) p=7: Lemma 3.5(a) (U, b1) witness per Definition 2.9, (Cl')  [t=31, |X|=31, |c|=14880 elements tested]
q= 31 done in 1.8 s
q= 32 PASS Lemma 3.2: C_PGammaL(U) = U  [|PGammaL|=163680, |C(U)|=32]
q= 32 PASS B = N_S(U) = U:H, |B| = q(q-1)/d, F(B) = U
q= 32 PASS (Cl') N_S(Sylow 2) soluble  [|P|=32, |N|=992]
q= 32 PASS (Cl') N_S(Sylow 3) soluble  [|P|=3, |N|=66]
q= 32 PASS (Cl') N_S(Sylow 11) soluble  [|P|=11, |N|=66]
q= 32 PASS (Cl') N_S(Sylow 31) soluble  [|P|=31, |N|=62]
q= 32 PASS (P2) t=3: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=33]
q= 32 PASS (P2) t=11: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q+1)/d  [|C_S(Q_t)|=33]
q= 32 PASS (P2) t=31: divides exactly one of (q-+1)/d, Q_t cyclic, C_S(Q_t) cyclic of order (q-1)/d  [|C_S(Q_t)|=31]
q= 32 PASS Lemma 3.1 r=5 k=1: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 5456, |C| = 6]
q= 32 PASS Lemma 3.1 r=5 k=2: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 5456, |C| = 6]
q= 32 PASS Lemma 3.1 r=5 k=3: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 5456, |C| = 6]
q= 32 PASS Lemma 3.1 r=5 k=4: C_PGL(phi0^k)=PGL(2,2); all elements of order r of the coset conjugate  [#order r = 5456, |C| = 6]
q= 32 PASS coset sigma^1 (r=5) p=2: Lemma 3.4 (Q_11, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=32736 elements tested]
q= 32 PASS coset sigma^1 (r=5) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^1 (r=5) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^1 (r=5) p=11: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^1 (r=5) p=31: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^1 (r=5) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^2 (r=5) p=2: Lemma 3.4 (Q_11, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=32736 elements tested]
q= 32 PASS coset sigma^2 (r=5) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^2 (r=5) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^2 (r=5) p=11: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^2 (r=5) p=31: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^2 (r=5) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^3 (r=5) p=2: Lemma 3.4 (Q_11, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=32736 elements tested]
q= 32 PASS coset sigma^3 (r=5) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^3 (r=5) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^3 (r=5) p=11: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^3 (r=5) p=31: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^3 (r=5) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^4 (r=5) p=2: Lemma 3.4 (Q_11, b1) witness per Definition 2.9, (Cl')  [t=11, |X|=11, |c|=32736 elements tested]
q= 32 PASS coset sigma^4 (r=5) p=3: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^4 (r=5) p=5: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^4 (r=5) p=11: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^4 (r=5) p=31: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 PASS coset sigma^4 (r=5) p=7: Lemma 3.3 (U, b2) witness per Definition 2.9, (Cl')  [|D|=992, |F*(D)|=32, 992 elements of c normalize X]
q= 32 done in 172.9 s
SUMMARY: 295 checks, 0 failed, q in [4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32]
ALL CHECKS PASSED
