C1 ok: W' = E:A_4 has order 96, contains T = SL(2,3) of order 24 (unique involution, |Z|=2, Q_8 normal).
C2 ok: Ghat = B:W' has order 768 (perm group on 16 points), stabiliser of mu in W' is T, |B:T| = 192.
C3 ok: character table of T computed; rational degree-2 irreducible psi: (order, value) = [(1, 2), (2, -2), (3, -1), (4, 0), (6, 1)]
C4 ok: chi = theta^Ghat is an integer-valued irreducible character of degree 8.
   number of subgroups of order 32 in the Sylow 2-subgroup: 1587
   number of subgroups of order 96 containing an order-32 subgroup of P: 24
C5 ok: for every subgroup U of index 8, chi|_U has no linear constituent => chi is NOT monomial.
C6 ok: 1 < C_3 < A_4 = V_4:C_3 < W' = E:A_4 < Ghat = B:W' with abelian kernels V_4, E, B.
ALL CHECKS PASSED: Ghat (order 768) is a semi-abelian group that is not an M-group.
Generators of Ghat as permutations of {0..15}:
   [(8, 9), (10, 11)]
   [(8, 9), (12, 13)]
   [(8, 9), (14, 15)]
   [(0, 1), (2, 3), (8, 10), (9, 11), (12, 14), (13, 15)]
   [(2, 3), (4, 5), (8, 14), (9, 15), (10, 12), (11, 13)]
   [(4, 5), (6, 7), (8, 10), (9, 11), (12, 14), (13, 15)]
   [(0, 2, 4), (1, 3, 5), (8, 10, 12), (9, 11, 13)]
   [(0, 2), (1, 3), (4, 6), (5, 7), (8, 14), (9, 15), (10, 12), (11, 13)]
