q=4 (l=2, f=2, |S|=60, |PGL|=60, |B|=12)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=6, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=10, soluble=True
  (P3) t=3: |C_S(Q_t)|=3 (torus 3), cyclic=True, |N_S(Q_t)|=6 -> True
  (P3) t=5: |C_S(Q_t)|=5 (torus 5), cyclic=True, |N_S(Q_t)|=10 -> True
  (P5) c=delta^0 sigma^1 r=2: #elements of order r in c=10, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^1 r=2 p=2: X=Q_5 |X|=5 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^1 r=2 p=3: X=U |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=5: X=U |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=7: X=U |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
q=5 (l=5, f=1, |S|=60, |PGL|=120, |B|=10)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=6, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=10, soluble=True
  (P4) #V4-classes=1, |N_S(V)|=[12], as expected=True
  (P3) t=3: |C_S(Q_t)|=3 (torus 3), cyclic=True, |N_S(Q_t)|=6 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=5 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=5 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=V4 |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=5 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=7 (l=7, f=1, |S|=168, |PGL|=336, |B|=21)
  Sylow 2: |P|=8, |N_S(P)|=8, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=6, soluble=True
  Sylow 7: |P|=7, |N_S(P)|=21, soluble=True
  (P4) #V4-classes=2, |N_S(V)|=[24, 24], as expected=True
  (P3) t=3: |C_S(Q_t)|=3 (torus 3), cyclic=True, |N_S(Q_t)|=6 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 p=7: EXCLUDED (q = 7 mod 8, diagonal, p = l) - not checked
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=8 (l=2, f=3, |S|=504, |PGL|=504, |B|=56)
  Sylow 2: |P|=8, |N_S(P)|=56, soluble=True
  Sylow 3: |P|=9, |N_S(P)|=18, soluble=True
  Sylow 7: |P|=7, |N_S(P)|=14, soluble=True
  (P3) t=3: |C_S(Q_t)|=9 (torus 9), cyclic=True, |N_S(Q_t)|=18 -> True
  (P3) t=7: |C_S(Q_t)|=7 (torus 7), cyclic=True, |N_S(Q_t)|=14 -> True
  (P5) c=delta^0 sigma^1 r=3: #elements of order r in c=84, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^1 r=3 p=2: X=Q_7 |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^1 r=3 p=3: X=U |X|=8 (b2): |D|=56, |F(D)|=8, |O_p(D)|=1, #beta in c normalizing X=56, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=3 p=7: X=U |X|=8 (b2): |D|=56, |F(D)|=8, |O_p(D)|=1, #beta in c normalizing X=56, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=3 p=5: X=U |X|=8 (b2): |D|=56, |F(D)|=8, |O_p(D)|=1, #beta in c normalizing X=56, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  (P5) c=delta^0 sigma^2 r=3: #elements of order r in c=84, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^2 r=3 p=2: X=Q_7 |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^2 r=3 p=3: X=U |X|=8 (b2): |D|=56, |F(D)|=8, |O_p(D)|=1, #beta in c normalizing X=56, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=3 p=7: X=U |X|=8 (b2): |D|=56, |F(D)|=8, |O_p(D)|=1, #beta in c normalizing X=56, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=3 p=5: X=U |X|=8 (b2): |D|=56, |F(D)|=8, |O_p(D)|=1, #beta in c normalizing X=56, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
q=9 (l=3, f=2, |S|=360, |PGL|=720, |B|=36)
  Sylow 2: |P|=8, |N_S(P)|=8, soluble=True
  Sylow 3: |P|=9, |N_S(P)|=36, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=10, soluble=True
  (P4) #V4-classes=2, |N_S(V)|=[24, 24], as expected=True
  (P3) t=5: |C_S(Q_t)|=5 (torus 5), cyclic=True, |N_S(Q_t)|=10 -> True
  (P5) c=delta^0 sigma^1 r=2: #elements of order r in c=30, one PGL-class=True, |C_PGL(phi0^k)|=24 (= 24) -> True
  c=delta^0 sigma^1 r=2 p=2: X=U |X|=9 (b2): |D|=36, |F(D)|=9, |O_p(D)|=1, #beta in c normalizing X=36, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=3: X=Q_5 |X|=5 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^1 r=2 p=5: X=U |X|=9 (b2): |D|=36, |F(D)|=9, |O_p(D)|=1, #beta in c normalizing X=36, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=7: X=U |X|=9 (b2): |D|=36, |F(D)|=9, |O_p(D)|=1, #beta in c normalizing X=36, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=9 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 p=3: EXCLUDED (q = 1 mod 8, diagonal, p = l) - not checked
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=9 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=9 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  (P5) c=delta^1 sigma^1 r=2: #elements of order r in c=0 (expected 0) -> True
  c=delta^1 sigma^1 r=2 p=2: X=U |X|=9 (b2): |D|=36, |F(D)|=9, |O_p(D)|=1, #beta in c normalizing X=36, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^1 r=2 p=3: X=Q_5 |X|=5 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^1 r=2 p=5: X=U |X|=9 (b2): |D|=36, |F(D)|=9, |O_p(D)|=1, #beta in c normalizing X=36, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^1 r=2 p=7: X=U |X|=9 (b2): |D|=36, |F(D)|=9, |O_p(D)|=1, #beta in c normalizing X=36, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
q=11 (l=11, f=1, |S|=660, |PGL|=1320, |B|=55)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=12, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=10, soluble=True
  Sylow 11: |P|=11, |N_S(P)|=55, soluble=True
  (P4) #V4-classes=1, |N_S(V)|=[12], as expected=True
  (P3) t=3: |C_S(Q_t)|=6 (torus 6), cyclic=True, |N_S(Q_t)|=12 -> True
  (P3) t=5: |C_S(Q_t)|=5 (torus 5), cyclic=True, |N_S(Q_t)|=10 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=11: X=V4 |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=13 (l=13, f=1, |S|=1092, |PGL|=2184, |B|=78)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=12, soluble=True
  Sylow 7: |P|=7, |N_S(P)|=14, soluble=True
  Sylow 13: |P|=13, |N_S(P)|=78, soluble=True
  (P4) #V4-classes=1, |N_S(V)|=[12], as expected=True
  (P3) t=3: |C_S(Q_t)|=6 (torus 6), cyclic=True, |N_S(Q_t)|=12 -> True
  (P3) t=7: |C_S(Q_t)|=7 (torus 7), cyclic=True, |N_S(Q_t)|=14 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=13 (b1): t=13 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=13 (b1): t=13 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=13 (b1): t=13 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=13: X=V4 |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=13 (b1): t=13 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=16 (l=2, f=4, |S|=4080, |PGL|=4080, |B|=240)
  Sylow 2: |P|=16, |N_S(P)|=240, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=30, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=30, soluble=True
  Sylow 17: |P|=17, |N_S(P)|=34, soluble=True
  (P3) t=3: |C_S(Q_t)|=15 (torus 15), cyclic=True, |N_S(Q_t)|=30 -> True
  (P3) t=5: |C_S(Q_t)|=15 (torus 15), cyclic=True, |N_S(Q_t)|=30 -> True
  (P3) t=17: |C_S(Q_t)|=17 (torus 17), cyclic=True, |N_S(Q_t)|=34 -> True
  (P5) c=delta^0 sigma^2 r=2: #elements of order r in c=68, one PGL-class=True, |C_PGL(phi0^k)|=60 (= 60) -> True
  c=delta^0 sigma^2 r=2 p=2: X=Q_17 |X|=17 (b1): t=17 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^2 r=2 p=3: X=U |X|=16 (b2): |D|=240, |F(D)|=16, |O_p(D)|=1, #beta in c normalizing X=240, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=2 p=5: X=U |X|=16 (b2): |D|=240, |F(D)|=16, |O_p(D)|=1, #beta in c normalizing X=240, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=2 p=17: X=U |X|=16 (b2): |D|=240, |F(D)|=16, |O_p(D)|=1, #beta in c normalizing X=240, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=2 p=7: X=U |X|=16 (b2): |D|=240, |F(D)|=16, |O_p(D)|=1, #beta in c normalizing X=240, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
q=17 (l=17, f=1, |S|=2448, |PGL|=4896, |B|=136)
  Sylow 2: |P|=16, |N_S(P)|=16, soluble=True
  Sylow 3: |P|=9, |N_S(P)|=18, soluble=True
  Sylow 17: |P|=17, |N_S(P)|=136, soluble=True
  (P4) #V4-classes=2, |N_S(V)|=[24, 24], as expected=True
  (P3) t=3: |C_S(Q_t)|=9 (torus 9), cyclic=True, |N_S(Q_t)|=18 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=17 (b1): t=17 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=17 (b1): t=17 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 p=17: EXCLUDED (q = 1 mod 8, diagonal, p = l) - not checked
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=17 (b1): t=17 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=19 (l=19, f=1, |S|=3420, |PGL|=6840, |B|=171)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=9, |N_S(P)|=18, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=20, soluble=True
  Sylow 19: |P|=19, |N_S(P)|=171, soluble=True
  (P4) #V4-classes=1, |N_S(V)|=[12], as expected=True
  (P3) t=3: |C_S(Q_t)|=9 (torus 9), cyclic=True, |N_S(Q_t)|=18 -> True
  (P3) t=5: |C_S(Q_t)|=10 (torus 10), cyclic=True, |N_S(Q_t)|=20 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=19 (b1): t=19 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=19 (b1): t=19 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=19 (b1): t=19 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=19: X=V4 |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=19 (b1): t=19 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=23 (l=23, f=1, |S|=6072, |PGL|=12144, |B|=253)
  Sylow 2: |P|=8, |N_S(P)|=8, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=24, soluble=True
  Sylow 11: |P|=11, |N_S(P)|=22, soluble=True
  Sylow 23: |P|=23, |N_S(P)|=253, soluble=True
  (P4) #V4-classes=2, |N_S(V)|=[24, 24], as expected=True
  (P3) t=3: |C_S(Q_t)|=12 (torus 12), cyclic=True, |N_S(Q_t)|=24 -> True
  (P3) t=11: |C_S(Q_t)|=11 (torus 11), cyclic=True, |N_S(Q_t)|=22 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=23 (b1): t=23 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=23 (b1): t=23 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=11: X=U |X|=23 (b1): t=23 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 p=23: EXCLUDED (q = 7 mod 8, diagonal, p = l) - not checked
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=23 (b1): t=23 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=25 (l=5, f=2, |S|=7800, |PGL|=15600, |B|=300)
  Sylow 2: |P|=8, |N_S(P)|=8, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=24, soluble=True
  Sylow 5: |P|=25, |N_S(P)|=300, soluble=True
  Sylow 13: |P|=13, |N_S(P)|=26, soluble=True
  (P4) #V4-classes=2, |N_S(V)|=[24, 24], as expected=True
  (P3) t=3: |C_S(Q_t)|=12 (torus 12), cyclic=True, |N_S(Q_t)|=24 -> True
  (P3) t=13: |C_S(Q_t)|=13 (torus 13), cyclic=True, |N_S(Q_t)|=26 -> True
  (P5) c=delta^0 sigma^1 r=2: #elements of order r in c=130, one PGL-class=True, |C_PGL(phi0^k)|=120 (= 120) -> True
  c=delta^0 sigma^1 r=2 p=2: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=3: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=5: X=Q_13 |X|=13 (b1): t=13 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^1 r=2 p=13: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=2 p=7: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=25 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=25 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 p=5: EXCLUDED (q = 1 mod 8, diagonal, p = l) - not checked
  c=delta^1 sigma^0 r=2 p=13: X=U |X|=25 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=25 (b1): t=5 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  (P5) c=delta^1 sigma^1 r=2: #elements of order r in c=0 (expected 0) -> True
  c=delta^1 sigma^1 r=2 p=2: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^1 r=2 p=3: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^1 r=2 p=5: X=Q_13 |X|=13 (b1): t=13 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^1 r=2 p=13: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^1 r=2 p=7: X=U |X|=25 (b2): |D|=300, |F(D)|=25, |O_p(D)|=1, #beta in c normalizing X=300, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
q=27 (l=3, f=3, |S|=9828, |PGL|=19656, |B|=351)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=27, |N_S(P)|=351, soluble=True
  Sylow 7: |P|=7, |N_S(P)|=28, soluble=True
  Sylow 13: |P|=13, |N_S(P)|=26, soluble=True
  (P4) #V4-classes=1, |N_S(V)|=[12], as expected=True
  (P3) t=7: |C_S(Q_t)|=14 (torus 14), cyclic=True, |N_S(Q_t)|=28 -> True
  (P3) t=13: |C_S(Q_t)|=13 (torus 13), cyclic=True, |N_S(Q_t)|=26 -> True
  (P5) c=delta^0 sigma^1 r=3: #elements of order r in c=819, one PGL-class=True, |C_PGL(phi0^k)|=24 (= 24) -> True
  c=delta^0 sigma^1 r=3 p=2: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=3 p=3: X=Q_7 |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^1 r=3 p=7: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=3 p=13: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=3 p=5: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  (P5) c=delta^0 sigma^2 r=3: #elements of order r in c=819, one PGL-class=True, |C_PGL(phi0^k)|=24 (= 24) -> True
  c=delta^0 sigma^2 r=3 p=2: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=3 p=3: X=Q_7 |X|=7 (b1): t=7 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^2 r=3 p=7: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=3 p=13: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=3 p=5: X=U |X|=27 (b2): |D|=351, |F(D)|=27, |O_p(D)|=1, #beta in c normalizing X=351, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=27 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=V4 |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=27 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=13: X=U |X|=27 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=27 (b1): t=3 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=29 (l=29, f=1, |S|=12180, |PGL|=24360, |B|=406)
  Sylow 2: |P|=4, |N_S(P)|=12, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=30, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=30, soluble=True
  Sylow 7: |P|=7, |N_S(P)|=28, soluble=True
  Sylow 29: |P|=29, |N_S(P)|=406, soluble=True
  (P4) #V4-classes=1, |N_S(V)|=[12], as expected=True
  (P3) t=3: |C_S(Q_t)|=15 (torus 15), cyclic=True, |N_S(Q_t)|=30 -> True
  (P3) t=5: |C_S(Q_t)|=15 (torus 15), cyclic=True, |N_S(Q_t)|=30 -> True
  (P3) t=7: |C_S(Q_t)|=14 (torus 14), cyclic=True, |N_S(Q_t)|=28 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=29 (b1): t=29 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=29 (b1): t=29 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=29 (b1): t=29 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=29 (b1): t=29 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=29: X=V4 |X|=4 (b2): |D|=12, |F(D)|=4, |O_p(D)|=1, #beta in c normalizing X=12, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^1 sigma^0 r=2 p=11: X=U |X|=29 (b1): t=29 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=31 (l=31, f=1, |S|=14880, |PGL|=29760, |B|=465)
  Sylow 2: |P|=32, |N_S(P)|=32, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=30, soluble=True
  Sylow 5: |P|=5, |N_S(P)|=30, soluble=True
  Sylow 31: |P|=31, |N_S(P)|=465, soluble=True
  (P4) #V4-classes=2, |N_S(V)|=[24, 24], as expected=True
  (P3) t=3: |C_S(Q_t)|=15 (torus 15), cyclic=True, |N_S(Q_t)|=30 -> True
  (P3) t=5: |C_S(Q_t)|=15 (torus 15), cyclic=True, |N_S(Q_t)|=30 -> True
  c=delta^1 sigma^0 r=2 p=2: X=U |X|=31 (b1): t=31 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=3: X=U |X|=31 (b1): t=31 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 r=2 p=5: X=U |X|=31 (b1): t=31 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^1 sigma^0 p=31: EXCLUDED (q = 7 mod 8, diagonal, p = l) - not checked
  c=delta^1 sigma^0 r=2 p=7: X=U |X|=31 (b1): t=31 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
q=32 (l=2, f=5, |S|=32736, |PGL|=32736, |B|=992)
  Sylow 2: |P|=32, |N_S(P)|=992, soluble=True
  Sylow 3: |P|=3, |N_S(P)|=66, soluble=True
  Sylow 11: |P|=11, |N_S(P)|=66, soluble=True
  Sylow 31: |P|=31, |N_S(P)|=62, soluble=True
  (P3) t=3: |C_S(Q_t)|=33 (torus 33), cyclic=True, |N_S(Q_t)|=66 -> True
  (P3) t=11: |C_S(Q_t)|=33 (torus 33), cyclic=True, |N_S(Q_t)|=66 -> True
  (P3) t=31: |C_S(Q_t)|=31 (torus 31), cyclic=True, |N_S(Q_t)|=62 -> True
  (P5) c=delta^0 sigma^1 r=5: #elements of order r in c=5456, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^1 r=5 p=2: X=Q_11 |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^1 r=5 p=3: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=5 p=5: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=5 p=11: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=5 p=31: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^1 r=5 p=7: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  (P5) c=delta^0 sigma^2 r=5: #elements of order r in c=5456, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^2 r=5 p=2: X=Q_11 |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^2 r=5 p=3: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=5 p=5: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=5 p=11: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=5 p=31: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^2 r=5 p=7: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  (P5) c=delta^0 sigma^3 r=5: #elements of order r in c=5456, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^3 r=5 p=2: X=Q_11 |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^3 r=5 p=3: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^3 r=5 p=5: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^3 r=5 p=11: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^3 r=5 p=31: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^3 r=5 p=7: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  (P5) c=delta^0 sigma^4 r=5: #elements of order r in c=5456, one PGL-class=True, |C_PGL(phi0^k)|=6 (= 6) -> True
  c=delta^0 sigma^4 r=5 p=2: X=Q_11 |X|=11 (b1): t=11 not in {p,r}=True, #beta in c centralizing X=0, class Aut-inv=True, N_S(X) soluble=True -> True
  c=delta^0 sigma^4 r=5 p=3: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^4 r=5 p=5: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^4 r=5 p=11: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^4 r=5 p=31: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
  c=delta^0 sigma^4 r=5 p=7: X=U |X|=32 (b2): |D|=992, |F(D)|=32, |O_p(D)|=1, #beta in c normalizing X=992, #acting trivially on D/F=0, class Aut-inv=True, D soluble=True -> True
MY PSL2 DEFINITION-LEVEL CHECKS PASSED
exit=0
