=== S5  |G| = 120  (degree 5)
  method A: |F(G)| = 1, |E(G)| = 60, |F*(G)| = 60, 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, 60, 120], h* = 2; F*_1 equal to method A: True
  exhaustive: 17 conjugacy classes of soluble subgroups (154 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 24
  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 6]
  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 24]
  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 24]
  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 24]
  question (h*(G) <= s(G)): OK;  time 0.0s
=== SL25  |G| = 120  (degree 24)
  method A: |F(G)| = 2, |E(G)| = 120, |F*(G)| = 120, h*(G/F*) = 0  =>  h*(G) = 1
  O_p(G) orders: p=2:2, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 120], h* = 1; F*_1 equal to method A: True
  exhaustive: 11 conjugacy classes of soluble subgroups (75 subgroups)
  s(G) = max h(K) over soluble K: 2 (exact), attained by class rep of order 24
  Theorem P, p = 3 (O_p(G)=1): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 24]
  Theorem P, p = 5 (O_p(G)=1): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 24]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 24]
  question (h*(G) <= s(G)): OK;  time 0.0s
=== GL25  |G| = 480  (degree 24)
  method A: |F(G)| = 4, |E(G)| = 120, |F*(G)| = 240, h*(G/F*) = 1  =>  h*(G) = 2
  O_p(G) orders: p=2:4, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 240, 480], h* = 2; F*_1 equal to method A: True
  exhaustive: 45 conjugacy classes of soluble subgroups (463 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 96
  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 96]
  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 96]
  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 96]
  question (h*(G) <= s(G)): OK;  time 0.1s
=== 2^4:A5perm  |G| = 960  (degree 10)
  method A: |F(G)| = 16, |E(G)| = 1, |F*(G)| = 16, h*(G/F*) = 1  =>  h*(G) = 2
  O_p(G) orders: p=2:16, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 16, 960], h* = 2; F*_1 equal to method A: True
  exhaustive: 61 conjugacy classes of soluble subgroups (1691 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 160
  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 160]
  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 160]
  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 160]
  question (h*(G) <= s(G)): OK;  time 0.1s
=== ASL24  |G| = 960  (degree 16)
  method A: |F(G)| = 16, |E(G)| = 1, |F*(G)| = 16, h*(G/F*) = 1  =>  h*(G) = 2
  O_p(G) orders: p=2:16, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 16, 960], h* = 2; F*_1 equal to method A: True
  exhaustive: 72 conjugacy classes of soluble subgroups (2266 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 160
  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 160]
  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 160]
  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 160]
  question (h*(G) <= s(G)): OK;  time 0.2s
=== 2^4:S5perm  |G| = 1920  (degree 10)
  method A: |F(G)| = 16, |E(G)| = 1, |F*(G)| = 16, h*(G/F*) = 2  =>  h*(G) = 3
  O_p(G) orders: p=2:16, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 16, 960, 1920], h* = 3; F*_1 equal to method A: True
  exhaustive: 193 conjugacy classes of soluble subgroups (6663 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 192
  Theorem P, p = 3 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 3 -> OK  [class rep of order 192]
  Theorem P, p = 5 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 3 -> OK  [class rep of order 192]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 3 -> OK  [class rep of order 192]
  question (h*(G) <= s(G)): OK;  time 0.7s
=== A5xS4  |G| = 1440  (degree 9)
  method A: |F(G)| = 4, |E(G)| = 60, |F*(G)| = 240, h*(G/F*) = 2  =>  h*(G) = 3
  O_p(G) orders: p=2:4, p=3:1, p=5:1
  method B (intersection of H.C_G(H/L) over chief factors): F*-series orders [1, 240, 720, 1440], h* = 3; F*_1 equal to method A: True
  exhaustive: 143 conjugacy classes of soluble subgroups (4268 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 144
  Theorem P, p = 3 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 3 -> OK  [class rep of order 144]
  Theorem P, p = 5 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 3 -> OK  [class rep of order 144]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 3 -> OK  [class rep of order 144]
  question (h*(G) <= s(G)): OK;  time 0.4s
=== ASL25  |G| = 3000  (degree 25)
  method A: |F(G)| = 25, |E(G)| = 1, |F*(G)| = 25, h*(G/F*) = 1  =>  h*(G) = 2
  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, 3000], h* = 2; F*_1 equal to method A: True
  exhaustive: 30 conjugacy classes of soluble subgroups (2202 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 600
  Theorem P, p = 2 (O_p(G)=1): max h(K/O_p(K)) = 3 (exact) vs h*(G) = 2 -> OK  [class rep of order 600]
  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 600]
  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 600]
  question (h*(G) <= s(G)): OK;  time 0.4s
=== A5xA5  |G| = 3600  (degree 10)
  method A: |F(G)| = 1, |E(G)| = 3600, |F*(G)| = 3600, h*(G/F*) = 0  =>  h*(G) = 1
  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], h* = 1; F*_1 equal to method A: True
  exhaustive: 94 conjugacy classes of soluble subgroups (8144 subgroups)
  s(G) = max h(K) over soluble K: 2 (exact), attained by class rep of order 6
  Theorem P, p = 2 (O_p(G)=1): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 6]
  Theorem P, p = 3 (O_p(G)=1): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 72]
  Theorem P, p = 5 (O_p(G)=1): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 6]
  Theorem P, p = 7 (O_p(G)=1, p does not divide |G|): max h(K/O_p(K)) = 2 (exact) vs h*(G) = 1 -> OK  [class rep of order 6]
  question (h*(G) <= s(G)): OK;  time 0.5s
=== A5wrC2  |G| = 7200  (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, 7200], h* = 2; F*_1 equal to method A: True
  exhaustive: 89 conjugacy classes of soluble subgroups (14786 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 288]
  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 0.9s
=== A5A5diag2  |G| = 7200  (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, 7200], h* = 2; F*_1 equal to method A: True
  exhaustive: 206 conjugacy classes of soluble subgroups (25812 subgroups)
  s(G) = max h(K) over soluble K: 3 (exact), attained by class rep of order 144
  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 6]
  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 144]
  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 144]
  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 144]
  question (h*(G) <= s(G)): OK;  time 2.2s
