== Brauer pair of order 729: certificate run ==
[P1] cocycle identity failures on V^3 (81^3 triples): 0
[P1] associativity failures over all 729^3 triples: 0
[P1] identity and two-sided inverses: OK
[P2] cocycle identity failures on V^3 (81^3 triples): 0
[P2] associativity failures over all 729^3 triples: 0
[P2] identity and two-sided inverses: OK
[P1] element order counts: {1: 1, 3: 80, 9: 648}; exponent = 9
[P1] |Z(G)| = 9; Z(G) == 0 x Z: True
[P1] set of commutators == Z: True
[P1] number of classes = 89; size counts = {1: 9, 9: 80}
[P1] every non-central class equals the coset gZ: True
[P1] beta(v, .) onto Z for every v != 0: True
[P1] [ (v,0), (w,0) ] = (0, beta(v,w)) for all v, w: True
[P1] (v,z)^3 = (0, phi(v)) for all elements: True
[P2] element order counts: {1: 1, 3: 80, 9: 648}; exponent = 9
[P2] |Z(G)| = 9; Z(G) == 0 x Z: True
[P2] set of commutators == Z: True
[P2] number of classes = 89; size counts = {1: 9, 9: 80}
[P2] every non-central class equals the coset gZ: True
[P2] beta(v, .) onto Z for every v != 0: True
[P2] [ (v,0), (w,0) ] = (0, beta(v,w)) for all v, w: True
[P2] (v,z)^3 = (0, phi(v)) for all elements: True
[P1] projection P -> V is a homomorphism (all pairs): True; spot check of linear characters failures = 0
[P2] projection P -> V is a homomorphism (all pairs): True; spot check of linear characters failures = 0
[P1] orthonormality of the 89 class functions (exact): failures = 0
[P2] orthonormality of the 89 class functions (exact): failures = 0
degrees: 81 of degree 1, 8 of degree 9; sum of squares = 729
bijection of classes preserves class sizes: True
bijection (classes, characters) preserves the full character table: True
power maps n = 1..18 well defined on classes and preserved: True
exponents: P1 = 9, P2 = 9
indicator sequences (nu_1..nu_18) agree under the bijection, for all 89 irreducibles: True
[P1] multiset of indicator sequences nu_1..nu_9 :
     80 x (0, 0, 1, 0, 0, 1, 0, 0, 1)
      8 x (0, 0, 0, 0, 0, 0, 0, 0, 9)
      1 x (1, 1, 1, 1, 1, 1, 1, 1, 1)
[P1] set S of values (j = 1..9): [0, 1, 9]
[P2] multiset of indicator sequences nu_1..nu_9 :
     80 x (0, 0, 1, 0, 0, 1, 0, 0, 1)
      8 x (0, 0, 0, 0, 0, 0, 0, 0, 9)
      1 x (1, 1, 1, 1, 1, 1, 1, 1, 1)
[P2] set S of values (j = 1..9): [0, 1, 9]
[P1] Omega = {g : g^3 = 1}: size 81, abelian: True
[P2] Omega = {g : g^3 = 1}: size 81, abelian: False
RESULT: P1, P2 are non-isomorphic groups of order 729 with the same character table, the same power maps, hence identical higher indicators of all irreducibles. ALL CHECKS PASSED.
