CONTROL 1 broken cocycle: group check = False | associativity fails at (1,1,1) (expected False)
D8 group check: True group ok
D8 element orders: {1: 1, 2: 5, 4: 2}
D8 classes: 5 size histogram: {1: 2, 2: 3}
D8 |Z(P)| = 2  #commutators = 2  commutators == centre: True
D8 Camina property (gP' in class of g for g not in P'): True
D8 #cubes = 8  cubes central: False
D8 |Omega| = 1  subgroup: True  abelian: True  commuting pairs in Omega: 1
D8 table exact in Z[omega]; exact orthonormal: True
D8 degrees: {1: 4, 2: 1}  sum d^2 = 8
Q8 group check: True group ok
Q8 element orders: {1: 1, 2: 1, 4: 6}
Q8 classes: 5 size histogram: {1: 2, 2: 3}
Q8 |Z(P)| = 2  #commutators = 2  commutators == centre: True
Q8 Camina property (gP' in class of g for g not in P'): True
Q8 #cubes = 8  cubes central: False
Q8 |Omega| = 1  subgroup: True  abelian: True  commuting pairs in Omega: 1
Q8 table exact in Z[omega]; exact orthonormal: True
Q8 degrees: {1: 4, 2: 1}  sum d^2 = 8
CONTROL 2 D8/Q8: class bijections preserving structure constants: 6 ; also preserving power maps: 0 (expected >0 and 0)
   D8 indicator seqs nu_1..nu_4: [(0, 1, 0, 1), (0, 1, 0, 1), (0, 1, 0, 1), (0, 1, 0, 2), (1, 1, 1, 1)]
   Q8 indicator seqs nu_1..nu_4: [(0, -1, 0, 2), (0, 1, 0, 1), (0, 1, 0, 1), (0, 1, 0, 1), (1, 1, 1, 1)]  differ: True
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
P3(rank-1 phi) group check: True group ok
P3(rank-1 phi) element orders: {1: 1, 3: 242, 9: 486}
P3(rank-1 phi) classes: 89 size histogram: {1: 9, 9: 80}
P3(rank-1 phi) |Z(P)| = 9  #commutators = 9  commutators == centre: True
P3(rank-1 phi) Camina property (gP' in class of g for g not in P'): True
P3(rank-1 phi) #cubes = 3  cubes central: True
P3(rank-1 phi) |Omega| = 243  subgroup: True  abelian: False  commuting pairs in Omega: 12393
P3(rank-1 phi) table exact in Z[omega]; exact orthonormal: True
P3(rank-1 phi) degrees: {1: 81, 9: 8}  sum d^2 = 729
CONTROL 3 P3 indicator multiset: [((0, 0, 1, 0, 0, 1, 0, 0, 1), 80), ((0, 0, 0, 0, 0, 0, 0, 0, 9), 6), ((0, 0, 9, 0, 0, 9, 0, 0, 9), 2), ((1, 1, 1, 1, 1, 1, 1, 1, 1), 1)]  differs from P1: True
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
CONTROL 4 tau(alpha=id): {'bijection': True, 'sizes': True, 'struct_consts': True, 'power_maps': False, 'table': True, 'indicators_under_sigma': True}
CONTROL 4 overall certificate with alpha=id: False (expected False)
CONTROL 5 corrupted table orthonormal: False (expected False)
