type | rank | N | classes | pos.roots/class | w0=-1 | Z(A) | [zeta] in H1 | d=ell(zeta) | gcd | transvection automorphisms | odd-graph check
A1 | 1 | 1 | 1 | [1] | True | <Delta> | [1] | 1 | 1 | Z/2 (lam=-2/d) | True
A2 | 2 | 3 | 1 | [3] | False | <Delta^2> | [6] | 6 | 6 | trivial | True
A3 | 3 | 6 | 1 | [6] | False | <Delta^2> | [12] | 12 | 12 | trivial | True
A4 | 4 | 10 | 1 | [10] | False | <Delta^2> | [20] | 20 | 20 | trivial | True
A5 | 5 | 15 | 1 | [15] | False | <Delta^2> | [30] | 30 | 30 | trivial | True
A6 | 6 | 21 | 1 | [21] | False | <Delta^2> | [42] | 42 | 42 | trivial | True
A7 | 7 | 28 | 1 | [28] | False | <Delta^2> | [56] | 56 | 56 | trivial | True
A8 | 8 | 36 | 1 | [36] | False | <Delta^2> | [72] | 72 | 72 | trivial | True
A9 | 9 | 45 | 1 | [45] | False | <Delta^2> | [90] | 90 | 90 | trivial | True
B2 | 2 | 4 | 2 | [2, 2] | True | <Delta> | [2, 2] | 4 | 2 | Z (lam.[zeta]=0) plus lam.[zeta]=-2 solutions (infinite dihedral) | True
B3 | 3 | 9 | 2 | [6, 3] | True | <Delta> | [6, 3] | 9 | 3 | Z (lam.[zeta]=0) | True
B4 | 4 | 16 | 2 | [12, 4] | True | <Delta> | [12, 4] | 16 | 4 | Z (lam.[zeta]=0) | True
B5 | 5 | 25 | 2 | [20, 5] | True | <Delta> | [20, 5] | 25 | 5 | Z (lam.[zeta]=0) | True
B6 | 6 | 36 | 2 | [6, 30] | True | <Delta> | [6, 30] | 36 | 6 | Z (lam.[zeta]=0) | True
B7 | 7 | 49 | 2 | [42, 7] | True | <Delta> | [42, 7] | 49 | 7 | Z (lam.[zeta]=0) | True
B8 | 8 | 64 | 2 | [8, 56] | True | <Delta> | [8, 56] | 64 | 8 | Z (lam.[zeta]=0) | True
B9 | 9 | 81 | 2 | [72, 9] | True | <Delta> | [72, 9] | 81 | 9 | Z (lam.[zeta]=0) | True
D4 | 4 | 12 | 1 | [12] | True | <Delta> | [12] | 12 | 12 | trivial | True
D5 | 5 | 20 | 1 | [20] | False | <Delta^2> | [40] | 40 | 40 | trivial | True
D6 | 6 | 30 | 1 | [30] | True | <Delta> | [30] | 30 | 30 | trivial | True
D7 | 7 | 42 | 1 | [42] | False | <Delta^2> | [84] | 84 | 84 | trivial | True
D8 | 8 | 56 | 1 | [56] | True | <Delta> | [56] | 56 | 56 | trivial | True
D9 | 9 | 72 | 1 | [72] | False | <Delta^2> | [144] | 144 | 144 | trivial | True
E6 | 6 | 36 | 1 | [36] | False | <Delta^2> | [72] | 72 | 72 | trivial | True
E7 | 7 | 63 | 1 | [63] | True | <Delta> | [63] | 63 | 63 | trivial | True
E8 | 8 | 120 | 1 | [120] | True | <Delta> | [120] | 120 | 120 | trivial | True
F4 | 4 | 24 | 2 | [12, 12] | True | <Delta> | [12, 12] | 24 | 12 | Z (lam.[zeta]=0) | True
G2 | 2 | 6 | 2 | [3, 3] | True | <Delta> | [3, 3] | 6 | 3 | Z (lam.[zeta]=0) | True
H3 | 3 | 15 | 1 | [15] | True | <Delta> | [15] | 15 | 15 | trivial | True
H4 | 4 | 60 | 1 | [60] | True | <Delta> | [60] | 60 | 60 | trivial | True
I2(3) | 2 | 3 | 1 | [3] | False | <Delta^2> | [6] | 6 | 6 | trivial | True
I2(4) | 2 | 4 | 2 | [2, 2] | True | <Delta> | [2, 2] | 4 | 2 | Z (lam.[zeta]=0) plus lam.[zeta]=-2 solutions (infinite dihedral) | True
I2(5) | 2 | 5 | 1 | [5] | False | <Delta^2> | [10] | 10 | 10 | trivial | True
I2(6) | 2 | 6 | 2 | [3, 3] | True | <Delta> | [3, 3] | 6 | 3 | Z (lam.[zeta]=0) | True
I2(7) | 2 | 7 | 1 | [7] | False | <Delta^2> | [14] | 14 | 14 | trivial | True
I2(8) | 2 | 8 | 2 | [4, 4] | True | <Delta> | [4, 4] | 8 | 4 | Z (lam.[zeta]=0) | True
I2(9) | 2 | 9 | 1 | [9] | False | <Delta^2> | [18] | 18 | 18 | trivial | True
I2(10) | 2 | 10 | 2 | [5, 5] | True | <Delta> | [5, 5] | 10 | 5 | Z (lam.[zeta]=0) | True
I2(11) | 2 | 11 | 1 | [11] | False | <Delta^2> | [22] | 22 | 22 | trivial | True
I2(12) | 2 | 12 | 2 | [6, 6] | True | <Delta> | [6, 6] | 12 | 6 | Z (lam.[zeta]=0) | True
I2(13) | 2 | 13 | 1 | [13] | False | <Delta^2> | [26] | 26 | 26 | trivial | True
I2(14) | 2 | 14 | 2 | [7, 7] | True | <Delta> | [7, 7] | 14 | 7 | Z (lam.[zeta]=0) | True
I2(15) | 2 | 15 | 1 | [15] | False | <Delta^2> | [30] | 30 | 30 | trivial | True
I2(16) | 2 | 16 | 2 | [8, 8] | True | <Delta> | [8, 8] | 16 | 8 | Z (lam.[zeta]=0) | True
N formulas A_n: n(n+1)/2, B_n: n^2, D_n: n(n-1) hold on all computed cases: True
exceptional (N, w0=-1): {'E6': (36, False), 'E7': (63, True), 'E8': (120, True), 'F4': (24, True), 'G2': (6, True), 'H3': (15, True), 'H4': (60, True)}
d = ell(zeta) >= 3 for every computed irreducible type of rank >= 2: True
types with H_1(A) = Z^2 (two classes of generators): ['B2', 'B3', 'B4', 'B5', 'B6', 'B7', 'B8', 'B9', 'F4', 'G2', 'I2(10)', 'I2(12)', 'I2(14)', 'I2(16)', 'I2(4)', 'I2(6)', 'I2(8)']
classes of simple reflections = components of the odd-edge graph (all cases): True
eta(ab) = ab and eta(ba) = ba in F(a,b): True
psi: a -> b^-1, b -> bab is a two-sided inverse of eta on F(a,b): True
eta^j(a) = (ab)^j a and eta^j(b) = a^-1 (ab)^-(j-1) in F(a,b) for j = 1..12: True
ALL CHECKS PASSED: True
