OK  tau is an involution
OK  sigma has order 3
OK  |U| = 64, |H| = 7
OK  |S| = 29120 = |Sz(8)|
OK  S is 2-transitive on 65 points
OK  sigma normalizes S and is not in S
OK  |<S, sigma>| = 87360 = 3|S|
OK  (Z2) B = N_S(U) has order 448 = q^2 (q-1)
OK  (Z2) B = U H (H complement of order 7)
OK  (Z2) H acts fixed-point-freely on U
OK  (Z2) C_B(U) <= U
OK  (Z2) F(B) = U (B soluble)
OK  (Z2) B/U cyclic of order 7: an element of B of order 7 exists and B = U<h>
OK  (Z2) O_7(B) = 1
OK  (Z2) sigma normalizes U, acts on H by h_lam -> h_{lam^2}
OK  U: exponent 4, |Z(U)| = 8
OK  (Z3) t=5: Sylow cyclic of order 5, C_S(Q_t) = torus of order 5, |N_S(Q_t)| = 20, soluble
OK  (Z3) t=5: every nontrivial element of the torus has centralizer the torus
OK  (Z3) t=7: Sylow cyclic of order 7, C_S(Q_t) = torus of order 7, |N_S(Q_t)| = 14, soluble
OK  (Z3) t=7: every nontrivial element of the torus has centralizer the torus
OK  (Z3) t=13: Sylow cyclic of order 13, C_S(Q_t) = torus of order 13, |N_S(Q_t)| = 52, soluble
OK  (Z3) t=13: every nontrivial element of the torus has centralizer the torus
OK  (Z4) k=1: elements of order 3 in S sigma^k = S-class of sigma^k (1456 elements), |C_S(sigma^k)| = 20
OK  (Z4) k=1: C_S(sigma^k) ~ Sz(2) = 5:4 (order 20, Sylow 5 normal, element of order 4)
OK  Lemma 3.1 k=1: #beta in c normalizing U = 448 (= |B|), none acts trivially on B/U (#trivial=0)
OK  Lemma 3.2 k=1: t = 7 not in {2, 3}, #beta in c centralizing Q_7 = 0 (expected 0)
OK  (Z4) k=2: elements of order 3 in S sigma^k = S-class of sigma^k (1456 elements), |C_S(sigma^k)| = 20
OK  (Z4) k=2: C_S(sigma^k) ~ Sz(2) = 5:4 (order 20, Sylow 5 normal, element of order 4)
OK  Lemma 3.1 k=2: #beta in c normalizing U = 448 (= |B|), none acts trivially on B/U (#trivial=0)
OK  Lemma 3.2 k=2: t = 7 not in {2, 3}, #beta in c centralizing Q_7 = 0 (expected 0)
OK  (Cl') N_S(U) = B soluble
OK  S-class of U is sigma-invariant
MY SZ(8) CHECKS PASSED
exit=0
