G1 group law and idempotents: done
G2 8T' (entry, lamps, power of t, coefficient):
   (1,1)  e_{} t^0  -1
   (1,1)  e_{} t^3  +4
   (1,1)  e_{0} t^0  -9
   (1,1)  e_{0} t^3  +4
   (1,1)  e_{0,1} t^0  -1
   (1,1)  e_{0,1,2} t^0  +1
   (1,1)  e_{0,2} t^0  +1
   (1,1)  e_{1} t^0  -1
   (1,1)  e_{1,2} t^0  +1
   (1,1)  e_{2} t^0  +1
   (1,2)  e_{} t^0  -4
   (1,2)  e_{0} t^0  +4
   (2,1)  e_{} t^3  +4
   (2,1)  e_{0} t^3  -4
   (2,2)  e_{} t^0  -5
   (2,2)  e_{0} t^0  -5
   (2,2)  e_{0} t^3  +8
   (2,2)  e_{0,1} t^0  +1
   (2,2)  e_{0,1,2} t^0  +1
   (2,2)  e_{0,2} t^0  -1
   (2,2)  e_{1} t^0  +1
   (2,2)  e_{1,2} t^0  +1
   (2,2)  e_{2} t^0  -1
   23 terms, largest |coefficient| = 9
G3 phi_K homomorphism: done (K = 1..9, 300 random pairs each)
G4 kernels of 8T' on Q[G_K]^2 directly in the group ring (|G_K| = K 2^K):
   K=1: dim ker = 3, d_K = 3/2 = 1.5000000, exact over Q, right = left  (0.0 s)
   K=2: dim ker = 4, d_K = 1/2 = 0.5000000, exact over Q, right = left  (0.0 s)
   K=3: dim ker = 30, d_K = 5/4 = 1.2500000, exact over Q, right = left  [Proposition 4.2: 5/4]  (0.0 s)
   K=4: dim ker = 26, d_K = 13/32 = 0.4062500, exact over Q, right = left  (0.9 s)
   K=5: dim ker = 62, d_K = 31/80 = 0.3875000, exact over Q, right = left  (11.3 s)
   K=6: dim ker = 198, d_K = 33/64 = 0.5156250, right = left  [Proposition 4.2: 33/64]  (0.1 s)
   K=7: dim ker = 331, d_K = 331/896 = 0.3694196, right = left  (2.8 s)
   K=8: dim ker = 746, d_K = 373/1024 = 0.3642578, right = left  (24.8 s)
   K=9: dim ker = 1818, d_K = 101/256 = 0.3945312  [Proposition 4.2: 101/256]  (13.8 s)
G5 block sum diag(8T',8T',8T',8T') in M_8(Z G): kernel = 4 x kernel of 8T' for K = 1, 2, 3
G6 Z/4 wr Z/1: dim ker = 6, normalized 3/2 (d_K = 3/2)
G6 Z/4 wr Z/2: dim ker = 16, normalized 1/2 (d_K = 1/2)
G6 Z/4 wr Z/3: dim ker = 240, normalized 5/4 (d_K = 5/4)
G6 Z/4 wr Z/4: dim ker = 416, normalized 13/32 (d_K = 13/32)
G7a relabelled U = 011 (same letter probabilities): d_3 = 5/4, d_6 = 33/64 (equal to those of T', as expected)
G7b alpha' = 1 + pi'_0 pi'_1 (P(U) = 1/4): d_3 = 9/8, d_6 = 53/128: differs from T' (detected)
G7c shift t^2 instead of t^3: d_3 = 7/12, d_6 = 27/64: differs from T' (detected)
checks: 3390, failures: 0
ALL CHECKS PASSED
time 53.8 s
