
==================== P1 ====================
PASS P1: Todd-Coxeter on the 21 relators gives 729 cosets (group order)
PASS P1: multiplication table is a Latin square
PASS P1: element 0 (empty word) is the identity
PASS P1: two-sided inverses
PASS P1: associativity on all 729^3 triples
PASS P1: pi: P -> V = F_3^4 is a homomorphism
PASS P1: word map onto P(phi) of the note is a bijection
PASS P1: it is a homomorphism on all 729^2 pairs, so the presented group is isomorphic to P(phi)
PASS P1: element orders {1: 1, 3: 80, 9: 648} (exponent 9)
PASS P1: |Z(P)| = 9 and the set of commutators equals Z(P)
PASS P1: P' = Z(P) (order 9), so P has class 2
PASS P1: Z(P) = <z1, z2> = {z1^k z2^l}
PASS P1: ker pi = Z(P)
PASS P1: 89 classes, sizes {1: 9, 9: 80}
PASS P1: Camina: the class of every g outside P' is gP'
PASS P1: the non-central classes are the fibres K_v = pi^{-1}(v), v != 0
PASS P1: {g^3} = Z(P), |mho_1| = 9
PASS P1: g^3 = z^{phi(pi g)} for all g (Proposition 4.1(c))
PASS P1: Omega = {g : g^3 = 1} is a subgroup of order 81
INFO P1: commuting pairs in Omega = 6561, |Z(Omega)| = 81
PASS P1: Lewis data |P:P'|, |P'|, |mho_1| = (81, 9, 9)
PASS P1: centralizer of each non-central g is pi^{-1}(F_9 pi(g)), abelian of order 81
PASS P1: D(P) has 7281 simple modules, sum of dim^2 = 531441 = 729^2
PASS P1: the 81 functions psi_xi are homomorphisms P -> mu_3 (all pairs)
PASS P1: A = pi^{-1}(F_9 (1,1)) has order 81, is abelian and contains Z(P)
INFO P1: y1 = x1 x3, y2 = x2 x4 have y1^3 = z^(1, 0), y2^3 = z^(0, 1)
PASS P1: x1^u1 x2^u2 is a transversal of A
PASS P1: mu_lambda, lambda=(0, 1), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(0, 1) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(0, 2), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(0, 2) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(1, 0), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(1, 0) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(1, 1), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(1, 1) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(1, 2), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(1, 2) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(2, 0), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(2, 0) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(2, 1), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(2, 1) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: mu_lambda, lambda=(2, 2), is a linear character of A extending omega^lambda
PASS P1: the 9x9 monomial matrices induced from mu_(2, 2) form a representation (all 729^2 pairs)
PASS P1: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P1: all 89 characters are class functions
PASS P1: exact orthonormality of the 89 characters in Z[zeta_9]
PASS P1: 89 orthonormal characters = number of classes, sum of squared degrees = 729: Irr(P) is complete
PASS P1: power maps g -> g^n, n = 0..18, are well defined on classes
PASS P1: nu_1..nu_18 of all 89 characters are integers; element sum = class formula
PASS P1: all 89^3 fusion coefficients are non-negative integers (exact)
INFO P1: indicator sequences (nu_1..nu_9): {(1, 1, 1, 1, 1, 1, 1, 1, 1): 1, (0, 0, 1, 0, 0, 1, 0, 0, 1): 80, (0, 0, 0, 0, 0, 0, 0, 0, 9): 8}
PASS P1: 1 x (1,...,1), 80 x (0,0,1,0,0,1,0,0,1), 8 x (0,...,0,9)
PASS P1: nu_{n+9} = nu_n for n = 1..9
PASS P1: literal set S = {0, 1, 9}
PASS P1: least n with g^n = 1 for all g is 9
PASS P1: products of Proposition 4.1(d) (psi chi = chi, chi_l chi_m = 9 chi_{l+m}, chi_l chi_-l = sum psi)

==================== P2 ====================
PASS P2: Todd-Coxeter on the 21 relators gives 729 cosets (group order)
PASS P2: multiplication table is a Latin square
PASS P2: element 0 (empty word) is the identity
PASS P2: two-sided inverses
PASS P2: associativity on all 729^3 triples
PASS P2: pi: P -> V = F_3^4 is a homomorphism
PASS P2: word map onto P(phi) of the note is a bijection
PASS P2: it is a homomorphism on all 729^2 pairs, so the presented group is isomorphic to P(phi)
PASS P2: element orders {1: 1, 3: 80, 9: 648} (exponent 9)
PASS P2: |Z(P)| = 9 and the set of commutators equals Z(P)
PASS P2: P' = Z(P) (order 9), so P has class 2
PASS P2: Z(P) = <z1, z2> = {z1^k z2^l}
PASS P2: ker pi = Z(P)
PASS P2: 89 classes, sizes {1: 9, 9: 80}
PASS P2: Camina: the class of every g outside P' is gP'
PASS P2: the non-central classes are the fibres K_v = pi^{-1}(v), v != 0
PASS P2: {g^3} = Z(P), |mho_1| = 9
PASS P2: g^3 = z^{phi(pi g)} for all g (Proposition 4.1(c))
PASS P2: Omega = {g : g^3 = 1} is a subgroup of order 81
INFO P2: commuting pairs in Omega = 2673, |Z(Omega)| = 9
PASS P2: Lewis data |P:P'|, |P'|, |mho_1| = (81, 9, 9)
PASS P2: centralizer of each non-central g is pi^{-1}(F_9 pi(g)), abelian of order 81
PASS P2: D(P) has 7281 simple modules, sum of dim^2 = 531441 = 729^2
PASS P2: the 81 functions psi_xi are homomorphisms P -> mu_3 (all pairs)
PASS P2: A = pi^{-1}(F_9 (1,1)) has order 81, is abelian and contains Z(P)
INFO P2: y1 = x1 x3, y2 = x2 x4 have y1^3 = z^(0, 0), y2^3 = z^(1, 1)
PASS P2: x1^u1 x2^u2 is a transversal of A
PASS P2: mu_lambda, lambda=(0, 1), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(0, 1) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(0, 2), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(0, 2) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(1, 0), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(1, 0) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(1, 1), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(1, 1) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(1, 2), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(1, 2) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(2, 0), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(2, 0) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(2, 1), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(2, 1) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: mu_lambda, lambda=(2, 2), is a linear character of A extending omega^lambda
PASS P2: the 9x9 monomial matrices induced from mu_(2, 2) form a representation (all 729^2 pairs)
PASS P2: its character is 9 omega^{lambda(z)} on Z(P) and 0 elsewhere (= chi_lambda of the note)
PASS P2: all 89 characters are class functions
PASS P2: exact orthonormality of the 89 characters in Z[zeta_9]
PASS P2: 89 orthonormal characters = number of classes, sum of squared degrees = 729: Irr(P) is complete
PASS P2: power maps g -> g^n, n = 0..18, are well defined on classes
PASS P2: nu_1..nu_18 of all 89 characters are integers; element sum = class formula
PASS P2: all 89^3 fusion coefficients are non-negative integers (exact)
INFO P2: indicator sequences (nu_1..nu_9): {(1, 1, 1, 1, 1, 1, 1, 1, 1): 1, (0, 0, 1, 0, 0, 1, 0, 0, 1): 80, (0, 0, 0, 0, 0, 0, 0, 0, 9): 8}
PASS P2: 1 x (1,...,1), 80 x (0,0,1,0,0,1,0,0,1), 8 x (0,...,0,9)
PASS P2: nu_{n+9} = nu_n for n = 1..9
PASS P2: literal set S = {0, 1, 9}
PASS P2: least n with g^n = 1 for all g is 9
PASS P2: products of Proposition 4.1(d) (psi chi = chi, chi_l chi_m = 9 chi_{l+m}, chi_l chi_-l = sum psi)

==================== the pair P1, P2 ====================
PASS tau is a bijection Cl(P1) -> Cl(P2)
PASS tau preserves class sizes
PASS tau(C^[n]) = tau(C)^[n] for n = 0..18 (hence for all n in Z, since g^9 = 1)
PASS sigma(chi)(tau(C)) = chi(C) for all 89 x 89 entries (exact): identical character tables
PASS row matching after tau finds the same sigma, uniquely
PASS negative control: with alpha replaced by the identity the power-map test fails
PASS negative control: with xi o alpha replaced by xi the character-table test fails
PASS nu_n(sigma(chi)) = nu_n(chi) for all chi and n = 1..18
PASS sigma preserves all 89^3 fusion coefficients
PASS commuting pairs in Omega: P1 6561 = 81^2 (abelian), P2 2673 = 9*81 + 72*27 (not abelian)
PASS Z(Omega(P2)) = Z(P2), of order 9
PASS in Omega(P2) every element outside Z(P2) has a centralizer of order 27
PASS Omega is characteristic, so P1 and P2 are not isomorphic

TOTAL checks: 133, failures: 0, time 10.0 s
