INFO C[Z/2]: sequences [(1, 1), (0, 1)]  C[Z/3]: sequences [(1, 1, 1), (0, 0, 1), (0, 0, 1)]
PASS literal sets coincide ({0,1}) although C[Z/2] = C^2 and C[Z/3] = C^3 have 2 and 3 simple modules
PASS Heis(3) has order 27, exponent 3 (1 + 26 elements of order 3) and is not abelian
PASS C^{Heis(3)}: Lambda^[n] = sum_{y^n = 1} delta_y by brute force over the iterated coproduct, n = 1..4
PASS C^G: Lambda = delta_1 satisfies f Lambda = eps(f) Lambda on the basis f = delta_x, and eps(Lambda) = Lambda(1) = 1
PASS exp C^{Heis(3)} = exp C[(Z/3)^3] = 3 (C[A] = C^{A^} for the dual)
INFO indexed indicator data, nu_1..nu_3: C^Heis {(1, 1, 1): 1, (0, 0, 1): 26}  C[A] {(1, 1, 1): 1, (0, 0, 1): 26}
PASS C^{Heis(3)} and C[(Z/3)^3]: identical indexed data 1 x (1,1,1,...), 26 x (0,0,1,...) for n = 1..18
PASS Grothendieck rings: Z[Heis(3)] (S_g x S_h = S_gh) is not commutative, that of Rep C[A] is
INFO order statistics of Z/8: ((1, 1), (2, 1), (4, 2), (8, 4))
INFO order statistics of Z/4xZ/2: ((1, 1), (2, 3), (4, 4))
INFO order statistics of D8: ((1, 1), (2, 5), (4, 2))
INFO order statistics of Q8: ((1, 1), (2, 1), (4, 6))
INFO order statistics of (Z/2)^3: ((1, 1), (2, 7))
PASS the five groups of order 8 have pairwise different numbers of elements of each order, so the five categories Vec_G (trivial cocycle) of rank 8 have pairwise different indicator data
INFO D8: nu_2 of the degree-2 character = 1
PASS D8: nu_2 of the degree-2 character is 1
INFO Q8: nu_2 of the degree-2 character = -1
PASS Q8: nu_2 of the degree-2 character is -1
PASS Q8' = {1, -1} has order 2, so dim C^{Q8/Q8'} = 4
PASS Q8 has 3 subgroups of order 4, all contain -1; restriction C^{Q8/Q8'} -> C^K has rank 2 for each

TOTAL checks: 12, failures: 0
