=== AGL25  |G| = 12000  (degree 25)
  method A: |F(G)| = 25, |E(G)| = 1, |F*(G)| = 25, h*(G/F*) = 2  =>  h*(G) = 3
  O_p(G) orders: p=2:1, p=3:1, p=5:25
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 25, 6000, 12000], h* = 3; F*_1 equal to method A: True
  exhaustive: 126 conjugacy classes of soluble subgroups (13562 subgroups)
  s(G) = max h(K) over soluble K: 4 (exact), attained by class rep of order 2400
  Theorem P, p = 2 (O_p(G)=1): max h(K/O_p(K)) = 4 (exact) vs h*(G) = 3 -> OK  [class rep of order 2400]
  Theorem P, p = 3 (O_p(G)=1): max h(K/O_p(K)) = 4 (exact) vs h*(G) = 3 -> OK  [class rep of order 2400]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 4 (exact) vs h*(G) = 3 -> OK  [class rep of order 2400]
  question (h*(G) <= s(G)): OK;  time 5.5s
=== 3^4:S5  |G| = 9720  (degree 81)
  method A: |F(G)| = 81, |E(G)| = 1, |F*(G)| = 81, h*(G/F*) = 2  =>  h*(G) = 3
  O_p(G) orders: p=2:1, p=3:81, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 81, 4860, 9720], h* = 3; F*_1 equal to method A: True
  exhaustive: 184 conjugacy classes of soluble subgroups (15868 subgroups)
  s(G) = max h(K) over soluble K: 4 (exact), attained by class rep of order 1944
  Theorem P, p = 2 (O_p(G)=1): max h(K/O_p(K)) = 4 (exact) vs h*(G) = 3 -> OK  [class rep of order 1944]
  Theorem P, p = 5 (O_p(G)=1): max h(K/O_p(K)) = 4 (exact) vs h*(G) = 3 -> OK  [class rep of order 1944]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 4 (exact) vs h*(G) = 3 -> OK  [class rep of order 1944]
  question (h*(G) <= s(G)): OK;  time 7.7s
=== S5wrC2  |G| = 28800  (degree 10)
  method A: |F(G)| = 1, |E(G)| = 3600, |F*(G)| = 3600, h*(G/F*) = 1  =>  h*(G) = 2
  O_p(G) orders: p=2:1, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 3600, 28800], h* = 2; F*_1 equal to method A: True
  exhaustive: 495 conjugacy classes of soluble subgroups (146194 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 288
  Theorem P, p = 2 (O_p(G)=1): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 2 -> OK  [class rep of order 48]
  Theorem P, p = 3 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 2 -> OK  [class rep of order 288]
  Theorem P, p = 5 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 2 -> OK  [class rep of order 288]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 2 -> OK  [class rep of order 288]
  question (h*(G) <= s(G)): OK;  time 14.3s
