group   k   log2|G|   d   |G^ab|  exp(G^ab)
Gamma   3      5     3       8   2
P       3      5     2       8   4
I       3      7     4      16   2
Gamma   4      8     3       8   2
P       4      8     2      16   8
I       4     10     4      16   2
Gamma   5     11     3       8   2
P       5     11     2      16   8
I       5     13     4      16   2
Gamma   6     14     3       8   2
P       6     14     2      16   8
I       6     16     4      16   2
PASS k=3: d(Gamma_k) = 3, Gamma_k^ab = C2^3
PASS k=3: d(P_k) = 2
PASS k=3: d(I_k) = 4, I_k^ab = C2^4
PASS k=3: |P_k| = |Gamma_k| = 2^(3k-4)
PASS k=4: d(Gamma_k) = 3, Gamma_k^ab = C2^3
PASS k=4: d(P_k) = 2
PASS k=4: d(I_k) = 4, I_k^ab = C2^4
PASS k=4: |P_k| = |Gamma_k| = 2^(3k-4)
PASS k=5: d(Gamma_k) = 3, Gamma_k^ab = C2^3
PASS k=5: d(P_k) = 2
PASS k=5: d(I_k) = 4, I_k^ab = C2^4
PASS k=5: |P_k| = |Gamma_k| = 2^(3k-4)
PASS k=6: d(Gamma_k) = 3, Gamma_k^ab = C2^3
PASS k=6: d(P_k) = 2
PASS k=6: d(I_k) = 4, I_k^ab = C2^4
PASS k=6: |P_k| = |Gamma_k| = 2^(3k-4)
PASS P_4^ab has order 16 and exponent 8
PASS T, w in P_4 (det 1, c even)
PASS <T, w> = P_4 (order 256)
PASS eps_1, eps_2, eps_3 are homomorphisms on Gamma_4 (all 65536 pairs)
eps on M_a, M_b, M_c: [(1, 0, 1), (1, 0, 0), (1, 1, 0)]
PASS values (1,0,1), (1,0,0), (1,1,0), invertible over F_2
PASS the four characters are homomorphisms on I_3 (all pairs)
PASS diag(-1,1), diag(5,1), T, L give the four unit vectors
G(B)_1: order 2^1, involutions 1
G(B)_2: order 2^3, involutions 5
G(B)_3: order 2^5, involutions 15
G(B)_4: order 2^7, involutions 43
PASS |G(B)_4| = 2^(3n-5)
PASS G(B)_4 has 43 involutions
G(B)_5: order 2^10, involutions 167
PASS |G(B)_5| = 2^(3n-5)
PASS G(B)_5 has 167 involutions
G(B)_6: order 2^13, involutions 647
PASS |G(B)_6| = 2^(3n-5)
PASS G(B)_6 has 647 involutions
PASS |G(B)_3| = 32
PASS d(G(B)_3) = 3 (automaton only)
PASS n=3: G(B)_n = kappa^-1 Gamma_n kappa (as permutation groups)
PASS n=3: kernel of Gamma_n on Z/2^n is [(1, 0, 0)]
PASS n=3: kernel of I_n on Z/2^n is {1, z_n}
PASS n=4: G(B)_n = kappa^-1 Gamma_n kappa (as permutation groups)
PASS n=4: kernel of Gamma_n on Z/2^n is [(1, 0, 0), (9, 0, 8)]
PASS n=4: P_n induces a group of order 2^(3n-5) on Z/2^n
PASS n=4: kernel of I_n on Z/2^n is {1, z_n}
PASS n=5: G(B)_n = kappa^-1 Gamma_n kappa (as permutation groups)
PASS n=5: kernel of Gamma_n on Z/2^n is [(1, 0, 0), (17, 0, 16)]
PASS n=5: P_n induces a group of order 2^(3n-5) on Z/2^n
PASS n=5: kernel of I_n on Z/2^n is {1, z_n}
PASS n=6: G(B)_n = kappa^-1 Gamma_n kappa (as permutation groups)
PASS n=6: kernel of Gamma_n on Z/2^n is [(1, 0, 0), (33, 0, 32)]
PASS n=6: P_n induces a group of order 2^(3n-5) on Z/2^n
PASS n=6: kernel of I_n on Z/2^n is {1, z_n}
n=3: |G(A)_n| = 2^4, tau in G(A)_n: False, |J_n|/|G(A)_n| = 2
n=4: |G(A)_n| = 2^7, tau in G(A)_n: False, |J_n|/|G(A)_n| = 2
PASS |G(A)_4| = 128 and tau not in G(A)_4
PASS n=4: [J_n : G(A)_n] = 2
n=5: |G(A)_n| = 2^10, tau in G(A)_n: False, |J_n|/|G(A)_n| = 2
PASS n=5: [J_n : G(A)_n] = 2
n=6: |G(A)_n| = 2^14, tau in G(A)_n: False, |J_n|/|G(A)_n| = 2
PASS n=6: [J_n : G(A)_n] = 2

SUMMARY r2_5_sylow: 0 failed
