P1 group check: True group ok
P1 element orders: {1: 1, 3: 80, 9: 648}
P1 classes: 89 size histogram: {1: 9, 9: 80}
P1 |Z(P)| = 9  #commutators = 9  commutators == centre: True
P1 Camina property (gP' in class of g for g not in P'): True
P1 #cubes = 9  cubes central: True
P1 |Omega| = 81  subgroup: True  abelian: True  commuting pairs in Omega: 6561
P1 table exact in Z[omega]; exact orthonormal: True
P1 degrees: {1: 81, 9: 8}  sum d^2 = 729
P2 group check: True group ok
P2 element orders: {1: 1, 3: 80, 9: 648}
P2 classes: 89 size histogram: {1: 9, 9: 80}
P2 |Z(P)| = 9  #commutators = 9  commutators == centre: True
P2 Camina property (gP' in class of g for g not in P'): True
P2 #cubes = 9  cubes central: True
P2 |Omega| = 81  subgroup: True  abelian: False  commuting pairs in Omega: 2673
P2 table exact in Z[omega]; exact orthonormal: True
P2 degrees: {1: 81, 9: 8}  sum d^2 = 729
P1 indicator sequences nu_1..nu_9 (multiset): [((0, 0, 1, 0, 0, 1, 0, 0, 1), 80), ((0, 0, 0, 0, 0, 0, 0, 0, 9), 8), ((1, 1, 1, 1, 1, 1, 1, 1, 1), 1)]
P1 nu_n periodic mod 9 for n<=18: True  set of values: [0, 1, 9]
P2 indicator sequences nu_1..nu_9 (multiset): [((0, 0, 1, 0, 0, 1, 0, 0, 1), 80), ((0, 0, 0, 0, 0, 0, 0, 0, 9), 8), ((1, 1, 1, 1, 1, 1, 1, 1, 1), 1)]
P2 nu_n periodic mod 9 for n<=18: True  set of values: [0, 1, 9]
tau(alpha=(a,c,b,d)): {'bijection': True, 'sizes': True, 'struct_consts': True, 'power_maps': True, 'table': True, 'indicators_under_sigma': True}
BRAUER PAIR CERTIFICATE (table + power maps under tau, sigma): True
fusion rules preserved by sigma: True
non-isomorphism invariant: Omega abelian in P1: True  in P2: False
