PASS S_4: 30 subgroups, h*(S_4) = h(S_4) = 3, s_2(S_4) = 2, s(S_4) = 3  [30 subgroups, h=3, s_2=2, s=3]
PASS N_PSL(2,7)(C_3) = S_3; delta chosen in PGL(2,7) - PSL(2,7) normalizing C_3; rho, tau normalize Q
PASS G/N = <images> in C_2 wr S_3 has order 24 and Fitting height 3 (S_4), so h*(G) = 1 + 3 = 4  [|G/N| = 24, h(G/N) = 3]
PASS K = N_G(Q) has order 216*24 = 5184, 7 does not divide |K|, h_7(K) = h(K) = 4 = h*(G)  [|K| = 5184, h(K) = 4]
SUMMARY: 4 checks, 0 failed
ALL CHECKS PASSED
