Part A: integer arithmetic
  A1 Lemma 3.3: 546 pairs (q, j) checked
  A2 Lemma 3.4: 731 field classes of prime order (l < 60, 2 <= f <= 30) checked
  A3 Lemma 4.3: 150 field classes of Sz(2^n), odd n <= 201, checked
  A4 Zsigmondy exceptions: 1003 pairs (l, n) checked
Part B: brute force in PGammaL(2,q)
  q=4
    Lemma 3.2: |C_PGammaL(U)| = 4 = |U|
    coset delta^0 sigma^1 (r=2): 12 elements normalize U, all move B/U; t=5, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=2, k=1: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 10 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 6, True), (5, 10, True)
  q=5
    Lemma 3.2: |C_PGammaL(U)| = 5 = |U|
    Lemma 3.5(b): Sylow 2-subgroup V is a Klein four-group, |N_S(V)| = 12, |N_PGL(V)| = 24, the 12 outer elements invert N_S(V)/V; one S-class of Klein four-groups (5 groups)
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 6, True), (5, 10, True)
  q=7
    Lemma 3.2: |C_PGammaL(U)| = 7 = |U|
    Remark 3.6: two S-classes of Klein four-groups (14 groups) interchanged by delta: True; Sylow 2-subgroup of order 8 self-normalizing: True; elements of delta centralize a Sylow t-subgroup for t = [3]
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 8, True), (3, 6, True), (7, 21, True)
  q=8
    Lemma 3.2: |C_PGammaL(U)| = 8 = |U|
    coset delta^0 sigma^1 (r=3): 56 elements normalize U, all move B/U; t=7, 0 elements centralize a Sylow t-subgroup
    coset delta^0 sigma^2 (r=3): 56 elements normalize U, all move B/U; t=7, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=3, k=1: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 84 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    Lemma 3.1, r=3, k=2: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 84 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 56, True), (3, 18, True), (7, 14, True)
  q=9
    Lemma 3.2: |C_PGammaL(U)| = 9 = |U|
    coset delta^0 sigma^1 (r=2): 36 elements normalize U, all move B/U; t=5, 0 elements centralize a Sylow t-subgroup
    coset delta^1 sigma^1 (r=2): 36 elements normalize U, all move B/U; t=5, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=2, k=1: |C_PGL(phi0^k)| = 24 = |PGL(2,3)|, 30 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,3)|
    Lemma 3.1: (PGL-PSL)phi0 contains 0 involutions
    Remark 3.6: two S-classes of Klein four-groups (30 groups) interchanged by delta: True; Sylow 2-subgroup of order 8 self-normalizing: True; elements of delta centralize a Sylow t-subgroup for t = [5]
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 8, True), (3, 36, True), (5, 10, True)
  q=11
    Lemma 3.2: |C_PGammaL(U)| = 11 = |U|
    Lemma 3.5(b): Sylow 2-subgroup V is a Klein four-group, |N_S(V)| = 12, |N_PGL(V)| = 24, the 12 outer elements invert N_S(V)/V; one S-class of Klein four-groups (55 groups)
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 12, True), (5, 10, True), (11, 55, True)
  q=13
    Lemma 3.2: |C_PGammaL(U)| = 13 = |U|
    Lemma 3.5(b): Sylow 2-subgroup V is a Klein four-group, |N_S(V)| = 12, |N_PGL(V)| = 24, the 12 outer elements invert N_S(V)/V; one S-class of Klein four-groups (91 groups)
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 12, True), (7, 14, True), (13, 78, True)
  q=16
    Lemma 3.2: |C_PGammaL(U)| = 16 = |U|
    coset delta^0 sigma^2 (r=2): 240 elements normalize U, all move B/U; t=17, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=2, k=1: |C_PGL(phi0^k)| = 60 = |PGL(2,4)|, 68 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,4)|
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 240, True), (3, 30, True), (5, 30, True), (17, 34, True)
  q=17
    Lemma 3.2: |C_PGammaL(U)| = 17 = |U|
    Remark 3.6: two S-classes of Klein four-groups (204 groups) interchanged by delta: True; Sylow 2-subgroup of order 16 self-normalizing: True; elements of delta centralize a Sylow t-subgroup for t = [3]
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 16, True), (3, 18, True), (17, 136, True)
  q=19
    Lemma 3.2: |C_PGammaL(U)| = 19 = |U|
    Lemma 3.5(b): Sylow 2-subgroup V is a Klein four-group, |N_S(V)| = 12, |N_PGL(V)| = 24, the 12 outer elements invert N_S(V)/V; one S-class of Klein four-groups (285 groups)
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 18, True), (5, 20, True), (19, 171, True)
  q=23
    Lemma 3.2: |C_PGammaL(U)| = 23 = |U|
    Remark 3.6: two S-classes of Klein four-groups (506 groups) interchanged by delta: True; Sylow 2-subgroup of order 8 self-normalizing: True; elements of delta centralize a Sylow t-subgroup for t = [3, 11]
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 8, True), (3, 24, True), (11, 22, True), (23, 253, True)
  q=25
    Lemma 3.2: |C_PGammaL(U)| = 25 = |U|
    coset delta^0 sigma^1 (r=2): 300 elements normalize U, all move B/U; t=13, 0 elements centralize a Sylow t-subgroup
    coset delta^1 sigma^1 (r=2): 300 elements normalize U, all move B/U; t=13, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=2, k=1: |C_PGL(phi0^k)| = 120 = |PGL(2,5)|, 130 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,5)|
    Lemma 3.1: (PGL-PSL)phi0 contains 0 involutions
    Remark 3.6: two S-classes of Klein four-groups (650 groups) interchanged by delta: True; Sylow 2-subgroup of order 8 self-normalizing: True; elements of delta centralize a Sylow t-subgroup for t = [3, 13]
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 8, True), (3, 24, True), (5, 300, True), (13, 26, True)
  q=27
    Lemma 3.2: |C_PGammaL(U)| = 27 = |U|
    coset delta^0 sigma^1 (r=3): 351 elements normalize U, all move B/U; t=7, 0 elements centralize a Sylow t-subgroup
    coset delta^0 sigma^2 (r=3): 351 elements normalize U, all move B/U; t=7, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=3, k=1: |C_PGL(phi0^k)| = 24 = |PGL(2,3)|, 819 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,3)|
    Lemma 3.1, r=3, k=2: |C_PGL(phi0^k)| = 24 = |PGL(2,3)|, 819 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,3)|
    Lemma 3.5(b): Sylow 2-subgroup V is a Klein four-group, |N_S(V)| = 12, |N_PGL(V)| = 24, the 12 outer elements invert N_S(V)/V; one S-class of Klein four-groups (819 groups)
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 351, True), (7, 28, True), (13, 26, True)
  q=29
    Lemma 3.2: |C_PGammaL(U)| = 29 = |U|
    Lemma 3.5(b): Sylow 2-subgroup V is a Klein four-group, |N_S(V)| = 12, |N_PGL(V)| = 24, the 12 outer elements invert N_S(V)/V; one S-class of Klein four-groups (1015 groups)
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 12, True), (3, 30, True), (5, 30, True), (7, 28, True), (29, 406, True)
  q=31
    Lemma 3.2: |C_PGammaL(U)| = 31 = |U|
    Remark 3.6: two S-classes of Klein four-groups (1240 groups) interchanged by delta: True; Sylow 2-subgroup of order 32 self-normalizing: True; elements of delta centralize a Sylow t-subgroup for t = [3, 5]
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 32, True), (3, 30, True), (5, 30, True), (31, 465, True)
  q=32
    Lemma 3.2: |C_PGammaL(U)| = 32 = |U|
    coset delta^0 sigma^1 (r=5): 992 elements normalize U, all move B/U; t=11, 0 elements centralize a Sylow t-subgroup
    coset delta^0 sigma^2 (r=5): 992 elements normalize U, all move B/U; t=11, 0 elements centralize a Sylow t-subgroup
    coset delta^0 sigma^3 (r=5): 992 elements normalize U, all move B/U; t=11, 0 elements centralize a Sylow t-subgroup
    coset delta^0 sigma^4 (r=5): 992 elements normalize U, all move B/U; t=11, 0 elements centralize a Sylow t-subgroup
    Lemma 3.1, r=5, k=1: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 5456 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    Lemma 3.1, r=5, k=2: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 5456 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    Lemma 3.1, r=5, k=3: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 5456 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    Lemma 3.1, r=5, k=4: |C_PGL(phi0^k)| = 6 = |PGL(2,2)|, 5456 elements of order r in PGL(2,q)phi0^k = |PGL(2,q)|/|PGL(2,2)|
    (Cl'): normalizers of Sylow t-subgroups (t, order, soluble): (2, 992, True), (3, 66, True), (11, 66, True), (31, 62, True)
Part C: small groups
  C1 S_4: 30 subgroups, h*(S_4) = h(S_4) = 3, s_2(S_4) = 2
  C2 S_5: normal subgroups 1, A_5, S_5 and A_5 perfect, so F*(S_5) = A_5 and h*(S_5) = 2; the point stabilizer S_4 has Fitting height 3
check_note: 2667 checks, 0 failures
ALL CHECKS PASSED
