(a) Lemma 3.7 directly (only chief factors below F*):
  Lemma 3.7 on S5          : chief factors below F* = ['NONAB60']; |Y| = 60, |F*| = 60 -> Y = F* (OK)
  Lemma 3.7 on SL25        : chief factors below F* = ['ab2', 'NONAB60']; |Y| = 120, |F*| = 120 -> Y = F* (OK)
  Lemma 3.7 on GL25        : chief factors below F* = ['ab2', 'ab2', 'NONAB60']; |Y| = 240, |F*| = 240 -> Y = F* (OK)
  Lemma 3.7 on 2^4:A5perm  : chief factors below F* = ['ab16']; |Y| = 16, |F*| = 16 -> Y = F* (OK)
  Lemma 3.7 on ASL24       : chief factors below F* = ['ab16']; |Y| = 16, |F*| = 16 -> Y = F* (OK)
  Lemma 3.7 on 2^4:S5perm  : chief factors below F* = ['ab16']; |Y| = 16, |F*| = 16 -> Y = F* (OK)
  Lemma 3.7 on A5xS4       : chief factors below F* = ['ab4', 'NONAB60']; |Y| = 240, |F*| = 240 -> Y = F* (OK)
  Lemma 3.7 on ASL25       : chief factors below F* = ['ab25']; |Y| = 25, |F*| = 25 -> Y = F* (OK)
  Lemma 3.7 on A5xA5       : chief factors below F* = ['NONAB60', 'NONAB60']; |Y| = 3600, |F*| = 3600 -> Y = F* (OK)
  Lemma 3.7 on A5wrC2      : chief factors below F* = ['NONAB3600']; |Y| = 3600, |F*| = 3600 -> Y = F* (OK)
  Lemma 3.7 on A5A5diag2   : chief factors below F* = ['NONAB60', 'NONAB60']; |Y| = 3600, |F*| = 3600 -> Y = F* (OK)
  Lemma 3.7 on AGL25       : chief factors below F* = ['ab25']; |Y| = 25, |F*| = 25 -> Y = F* (OK)
  Lemma 3.7 on 3^4:S5      : chief factors below F* = ['ab81']; |Y| = 81, |F*| = 81 -> Y = F* (OK)
(c) deliberately wrong inputs
   missing component in A5 wr C2: rejected (C_G(F E0) not inside F E0: components missing)
   diagonal A5 offered as component of A5 x A5: rejected (component not subnormal)
   fitting_height(A5) refused (not soluble)
   known values: h(S4) = 3 (3), h(GL(2,3)) = 3 (3), h(SL(2,3)) = 2 (2), h_2(S4) = 2 (2), |GL(2,3)| = 48
   h*(A5 x C2) = 1 (quasinilpotent, so 1; differs from h*(S5) = 2)
(b) wreath products checked through their structure (G itself not enumerated)
   C_{S5}(A5) order: 1
   A5 wr S4: h*(G) = 1 + h(S4) = 4
   K = S3 wr S4, |K| = 31104: h(K) = 4, h_2 = 4, h_3 = 3, h_5 = 4  (0.3s)
   K = D10 wr S4, |K| = 240000: h(K) = 4, h_2 = 4, h_3 = 4, h_5 = 3  (3.3s)
   K = A4 wr S4, |K| = 497664: h(K) = 5, h_2 = 4, h_3 = 5, h_5 = 5  (5.7s)
   S5 wr S3: |C2 wr S3| = 48, h(C2 wr S3) = 3, so h*(G) = 4
   K = S4 wr S3, |K| = 82944: h(K) = 5, h_2 = 4, h_3 = 5, h_5 = 5  (1.2s)
   K = F20 wr S3, |K| = 48000: h(K) = 4, h_2 = 4, h_3 = 4, h_5 = 3  (0.4s)
   K = S3 wr S3, |K| = 1296: h(K) = 4, h_2 = 4, h_3 = 3, h_5 = 4  (0.0s)
