[    0.4s] S0: group Ghat built: |Ghat| = 768 ; multiplication table done
[    0.4s] S1 ok: chain 1 < C3 < A4 = V4:C3 < W = E:A4 < Ghat = B:W, kernels C3, V4, E, B abelian; |Ghat|=768, |W|=96, |A4|=12
[    0.4s] S1 ok: W|_{0..7} is an index-2 subgroup of C2 wr A4 <= S8
[    0.4s] S2 ok: mu is a linear character of B with stabiliser T of order 24 in W; T' has index 3 in T; T has a unique involution; orbit sizes of W on Irr(B): 1, 3, 4
[    0.4s] S2 ok: subgroup orders of T are [1, 2, 3, 3, 3, 3, 4, 4, 4, 6, 6, 6, 6, 8, 24] -> no subgroup of index 2
[    0.4s] S2 ok: explicit isomorphisms T -> binary tetrahedral group (quaternions) and T -> SL(2,F_3) found
[    0.4s] S3 ok: psi (deg 2, rational, values by element order [(1, 2), (2, -2), (3, -1), (4, 0), (6, 1)] ) is irreducible; psi3 (deg 3) irreducible; T/T' = C3 with linear characters omega^k
[    0.7s] S4 ok: chi = (mu~ psi)^Ghat is an irreducible integer-valued character of degree 8; chi|_B = 2*(sum of the W-orbit of mu); control characters chi4, chi4b (deg 4), chi12 (deg 12), chi8b (deg 8) irreducible
[    0.7s] S5: subgroups of P of order 2: 87
[    0.7s] S5: subgroups of P of order 4: 563
[    0.7s] S5: subgroups of P of order 8: 1451
[    0.9s] S5: subgroups of P of order 16: 1947
[    1.5s] S5: subgroups of P of order 32: 1587
[    2.8s] S5: subgroups of P of order 64: 651
[    3.6s] S5: subgroups of P of order 128: 63
[    3.7s] S5: subgroups of P of order 256: 1
[    3.7s] S5: elements of order 3 in Ghat: 128 (Sylow 3-subgroups: 64)
[    3.8s] S5 (method E): subgroups of Ghat of order 96: 24, of order 192: 84, of order 384: 0
[    3.9s] S6: subgroups of W by order: {1: 1, 2: 19, 3: 16, 4: 39, 6: 16, 8: 35, 12: 12, 16: 15, 24: 20, 32: 1, 96: 1}
[    3.9s] S6 (method B): subgroups of order 96 found: 24 ; by (|Ubar|, |B0|): {(12, 8): 12, (24, 4): 4, (96, 1): 8}
[    3.9s] S6 ok: the two enumerations of all subgroups of order 96 agree exactly
[    3.9s] S7: chi (deg 8): subgroups of index 8 on which chi|_U has a linear constituent: 0 of 24
[    3.9s] S7 ok: chi is NOT monomial (all twisted sums vanish on all 24 subgroups of index 8)
[    3.9s] S7 ok: chi8b = (mu~ psi omega)^Ghat is not monomial either
[    4.0s] S7 ok: control chi4 (deg 4): 12 of 84 index-4 subgroups carry a linear constituent (e.g. multiplicities [1])
[    4.0s] S7 ok: control chi4b (deg 4): 12 of 84 index-4 subgroups carry a linear constituent
[    4.0s] S7 ok: control chi12 (deg 12): 15 of 651 index-12 subgroups carry a linear constituent; B:Q8 is one of them
[    4.0s] S8: Ghat has 35 conjugacy classes
[    4.1s] S8: irreducible character degrees of Ghat: [1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 8, 8, 8, 12, 12]
[    4.1s] S8 ok: exact characters chi, chi8b, chi4, chi4b, chi12 coincide with table rows 22, 15, 29, 7, 33
[    4.3s] S8: monomiality of every irreducible character of Ghat (row, degree, #index-d subgroups with a linear constituent / total):
      row  0  degree  1     1 /    1  MONOMIAL
      row  1  degree  1     1 /    1  MONOMIAL
      row  2  degree  1     1 /    1  MONOMIAL
      row  3  degree  3     1 /    1  MONOMIAL
      row  6  degree  3     1 /    1  MONOMIAL
      row 10  degree  3     1 /    1  MONOMIAL
      row 11  degree  3     1 /    1  MONOMIAL
      row 12  degree  3     1 /    1  MONOMIAL
      row 13  degree  3     1 /    1  MONOMIAL
      row 16  degree  3     1 /    1  MONOMIAL
      row 17  degree  3     1 /    1  MONOMIAL
      row 18  degree  3     1 /    1  MONOMIAL
      row 19  degree  3     1 /    1  MONOMIAL
      row 20  degree  3     1 /    1  MONOMIAL
      row 21  degree  3     1 /    1  MONOMIAL
      row 23  degree  3     1 /    1  MONOMIAL
      row 24  degree  3     1 /    1  MONOMIAL
      row 25  degree  3     1 /    1  MONOMIAL
      row 26  degree  3     1 /    1  MONOMIAL
      row 27  degree  3     1 /    1  MONOMIAL
      row 30  degree  3     1 /    1  MONOMIAL
      row 31  degree  3     1 /    1  MONOMIAL
      row 32  degree  3     1 /    1  MONOMIAL
      row 34  degree  3     1 /    1  MONOMIAL
      row  4  degree  4    12 /   84  MONOMIAL
      row  5  degree  4    12 /   84  MONOMIAL
      row  7  degree  4    12 /   84  MONOMIAL
      row  8  degree  4    12 /   84  MONOMIAL
      row  9  degree  4    12 /   84  MONOMIAL
      row 29  degree  4    12 /   84  MONOMIAL
      row 14  degree  8     0 /   24  NOT monomial
      row 15  degree  8     0 /   24  NOT monomial
      row 22  degree  8     0 /   24  NOT monomial
      row 28  degree 12    15 /  651  MONOMIAL
      row 33  degree 12    15 /  651  MONOMIAL
[    4.3s] S8 ok: exactly the three irreducible characters of degree 8 (chi and its two twists by the linear characters of Ghat/P) are non-monomial; all other irreducibles are monomial

SUMMARY
  |Ghat| = 768, semi-abelian chain 1 < C3 < A4 < W < Ghat verified (abelian kernels C3, V4, C2^3, C2^3)
  W_mu = T, |T| = 24, T = SL(2,3) (explicit isomorphism), [T:T'] = 3, no subgroup of index 2
  chi = (mu~ psi)^Ghat: irreducible, degree 8, integer valued
  subgroups of P of order 2^k: {1: 1, 2: 87, 4: 563, 8: 1451, 16: 1947, 32: 1587, 64: 651, 128: 63, 256: 1}
  subgroups of Ghat of order 96: 24 (method E) = 24 (method B), identical sets
  chi|_U has no linear constituent for any of them  =>  chi is not monomial  =>  Ghat is not an M-group
  full table: 35 irreducibles, degrees [1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 8, 8, 8, 12, 12]; non-monomial ones: [8, 8, 8]
  total time 4.3s
AUDIT CHECKS ALL PASSED
