P2 = PGammaL(3,4): |P2| = 120960, M = 384; P1 = PGL(3,4): 100 classes of subgroups
classes of subgroups of P1 of order > 192: 15 (orders [216, 240, 240, 288, 288, 360, 480, 480, 576, 960, 960, 2880, 2880, 20160, 60480])
  good class of P1: |S0| = 240, good at [(1, 5)] (orbit length, image order); |N_P2(S0)| = 960; extensions S = S0.2 not inside P1: 0, good ones: 0
  good class of P1: |S0| = 240, good at [(16, 5)] (orbit length, image order); |N_P2(S0)| = 960; extensions S = S0.2 not inside P1: 0, good ones: 0
good classes of P1 of order > 192: 2
RESULT: largest order of a good subgroup of P2 found among P1-subgroups of order > 192 and their extensions: 240;  M = 384;  hence M_G(PGammaL(3,4)) = 384  (strict = False)  [2s]
