== (1) cycles
cycle C_3                n=    3 d= 2 tau(0)=    0.6667 sup=    1.9998 at p=9.999e-01 sup/tau0= 2.9997 sup/max(tau0,d)= 0.9999 unimodal(inc then dec)=True  max 2Var=   0.5000 (log_d n)^2=  2.51
cycle C_4                n=    4 d= 2 tau(0)=    1.0000 sup=    2.0000 at p=9.999e-01 sup/tau0= 2.0000 sup/max(tau0,d)= 1.0000 unimodal(inc then dec)=True  max 2Var=   1.0361 (log_d n)^2=  4.00
cycle C_5                n=    5 d= 2 tau(0)=    1.4472 sup=    2.4499 at p=7.308e-01 sup/tau0= 1.6928 sup/max(tau0,d)= 1.2249 unimodal(inc then dec)=True  max 2Var=   1.2132 (log_d n)^2=  5.39
cycle C_6                n=    6 d= 2 tau(0)=    2.0000 sup=    2.9469 at p=4.800e-01 sup/tau0= 1.4734 sup/max(tau0,d)= 1.4734 unimodal(inc then dec)=True  max 2Var=   1.8959 (log_d n)^2=  6.68
cycle C_7                n=    7 d= 2 tau(0)=    2.6560 sup=    3.6530 at p=3.478e-01 sup/tau0= 1.3754 sup/max(tau0,d)= 1.3754 unimodal(inc then dec)=True  max 2Var=   2.2318 (log_d n)^2=  7.88
cycle C_8                n=    8 d= 2 tau(0)=    3.4142 sup=    4.4582 at p=2.400e-01 sup/tau0= 1.3058 sup/max(tau0,d)= 1.3058 unimodal(inc then dec)=True  max 2Var=   3.0840 (log_d n)^2=  9.00
cycle C_9                n=    9 d= 2 tau(0)=    4.2743 sup=    5.3927 at p=1.800e-01 sup/tau0= 1.2617 sup/max(tau0,d)= 1.2617 unimodal(inc then dec)=True  max 2Var=   3.5837 (log_d n)^2= 10.05
cycle C_10               n=   10 d= 2 tau(0)=    5.2361 sup=    6.4323 at p=1.383e-01 sup/tau0= 1.2285 sup/max(tau0,d)= 1.2285 unimodal(inc then dec)=True  max 2Var=   4.6079 (log_d n)^2= 11.04
cycle C_12               n=   12 d= 2 tau(0)=    7.4641 sup=    8.8494 at p=8.359e-02 sup/tau0= 1.1856 sup/max(tau0,d)= 1.1856 unimodal(inc then dec)=True  max 2Var=   6.4695 (log_d n)^2= 12.85
cycle C_16               n=   16 d= 2 tau(0)=   13.1371 sup=   14.9795 at p=3.817e-02 sup/tau0= 1.1402 sup/max(tau0,d)= 1.1402 unimodal(inc then dec)=True  max 2Var=  11.2078 (log_d n)^2= 16.00
cycle C_24               n=   24 d= 2 tau(0)=   29.3477 sup=   32.3698 at p=1.377e-02 sup/tau0= 1.1030 sup/max(tau0,d)= 1.1030 unimodal(inc then dec)=True  max 2Var=  24.7454 (log_d n)^2= 21.02
cycle C_32               n=   32 d= 2 tau(0)=   52.0434 sup=   56.5709 at p=7.225e-03 sup/tau0= 1.0870 sup/max(tau0,d)= 1.0870 unimodal(inc then dec)=True  max 2Var=  43.6984 (log_d n)^2= 25.00
cycle C_48               n=   48 d= 2 tau(0)=  116.8888 sup=  125.3933 at p=2.794e-03 sup/tau0= 1.0728 sup/max(tau0,d)= 1.0728 unimodal(inc then dec)=True  max 2Var=  97.8492 (log_d n)^2= 31.19
cycle C_64               n=   64 d= 2 tau(0)=  207.6725 sup=  221.4218 at p=1.466e-03 sup/tau0= 1.0662 sup/max(tau0,d)= 1.0662 unimodal(inc then dec)=True  max 2Var= 173.6602 (log_d n)^2= 36.00
cycle C_96               n=   96 d= 2 tau(0)=  467.0547 sup=  495.0898 at p=6.078e-04 sup/tau0= 1.0600 sup/max(tau0,d)= 1.0600 unimodal(inc then dec)=True  max 2Var= 390.2631 (log_d n)^2= 43.36
cycle C_128              n=  128 d= 2 tau(0)=  830.1898 sup=  877.5361 at p=3.189e-04 sup/tau0= 1.0570 sup/max(tau0,d)= 1.0570 unimodal(inc then dec)=True  max 2Var= 693.5074 (log_d n)^2= 49.00
cycle C_192              n=  192 d= 2 tau(0)= 1867.7187 sup= 1968.9573 at p=1.417e-04 sup/tau0= 1.0542 sup/max(tau0,d)= 1.0542 unimodal(inc then dec)=True  max 2Var=1559.9217 (log_d n)^2= 57.53
cycle C_256              n=  256 d= 2 tau(0)= 3320.2592 sup= 3495.5538 at p=7.973e-05 sup/tau0= 1.0528 sup/max(tau0,d)= 1.0528 unimodal(inc then dec)=True  max 2Var=2772.9017 (log_d n)^2= 64.00

== (2) SL_2(Z_q), generators [[1,+-1],[0,1]], [[1,0],[+-1,1]]
SL_2(Z_3)                n=   24 d= 4 tau(0)=    3.1547 sup=    6.5971 at p=4.000e-01 sup/tau0= 2.0912 sup/max(tau0,d)= 1.6493 unimodal(inc then dec)=True  max 2Var=   2.4029 (log_d n)^2=  5.26
   elapsed 1s
SL_2(Z_5)                n=  120 d= 4 tau(0)=    5.2361 sup=   12.0383 at p=2.000e-01 sup/tau0= 2.2991 sup/max(tau0,d)= 2.2991 unimodal(inc then dec)=True  max 2Var=   5.1131 (log_d n)^2= 11.93
   elapsed 1s
SL_2(Z_7)                n=  336 d= 4 tau(0)=    6.8284 sup=   17.9847 at p=1.200e-01 sup/tau0= 2.6338 sup/max(tau0,d)= 2.6338 unimodal(inc then dec)=True  max 2Var=   7.6724 (log_d n)^2= 17.61
   elapsed 1s
SL_2(Z_11)               n= 1320 d= 4 tau(0)=   10.4721 sup=   30.4817 at p=6.310e-02 sup/tau0= 2.9107 sup/max(tau0,d)= 2.9107 unimodal(inc then dec)=True  max 2Var=  12.7363 (log_d n)^2= 26.87
   elapsed 5s
SL_2(Z_13)               n= 2184 d= 4 tau(0)=   12.3127 sup=   36.7605 at p=6.310e-02 sup/tau0= 2.9856 sup/max(tau0,d)= 2.9856 unimodal(inc then dec)=True  max 2Var=  15.2631 (log_d n)^2= 30.76
   elapsed 22s

== (3) symmetric groups
S_3 transpositions       n=    6 d= 3 tau(0)=    1.0000 sup=    3.0000 at p=9.990e-01 sup/tau0= 3.0000 sup/max(tau0,d)= 1.0000 unimodal(inc then dec)=True  max 2Var=   1.1022 (log_d n)^2=  2.66
S_4 transpositions       n=   24 d= 6 tau(0)=    1.5000 sup=    6.2665 at p=7.400e-01 sup/tau0= 4.1777 sup/max(tau0,d)= 1.0444 unimodal(inc then dec)=True  max 2Var=   1.9661 (log_d n)^2=  3.15
S_5 transpositions       n=  120 d=10 tau(0)=    2.0000 sup=   11.4806 at p=5.200e-01 sup/tau0= 5.7403 sup/max(tau0,d)= 1.1481 unimodal(inc then dec)=True  max 2Var=   3.1212 (log_d n)^2=  4.32
S_6 transpositions       n=  720 d=15 tau(0)=    2.5000 sup=   19.4123 at p=3.400e-01 sup/tau0= 7.7649 sup/max(tau0,d)= 1.2942 unimodal(inc then dec)=True  max 2Var=   4.5933 (log_d n)^2=  5.90
S_4 adjacent transp.     n=   24 d= 3 tau(0)=    5.1213 sup=    7.5824 at p=1.800e-01 sup/tau0= 1.4805 sup/max(tau0,d)= 1.4805 unimodal(inc then dec)=True  max 2Var=   4.8021 (log_d n)^2=  8.37
S_5 adjacent transp.     n=  120 d= 4 tau(0)=   10.4721 sup=   16.1579 at p=7.279e-02 sup/tau0= 1.5429 sup/max(tau0,d)= 1.5429 unimodal(inc then dec)=True  max 2Var=   9.9827 (log_d n)^2= 11.93
S_6 adjacent transp.     n=  720 d= 5 tau(0)=   18.6603 sup=   30.4511 at p=3.857e-02 sup/tau0= 1.6319 sup/max(tau0,d)= 1.6319 unimodal(inc then dec)=True  max 2Var=  18.4292 (log_d n)^2= 16.71
S_4 star transp.         n=   24 d= 3 tau(0)=    3.0000 sup=    6.3869 at p=3.200e-01 sup/tau0= 2.1290 sup/max(tau0,d)= 2.1290 unimodal(inc then dec)=True  max 2Var=   3.2895 (log_d n)^2=  8.37
S_5 star transp.         n=  120 d= 4 tau(0)=    4.0000 sup=   11.7532 at p=2.200e-01 sup/tau0= 2.9383 sup/max(tau0,d)= 2.9383 unimodal(inc then dec)=True  max 2Var=   5.1236 (log_d n)^2= 11.93
S_6 star transp.         n=  720 d= 5 tau(0)=    5.0000 sup=   19.7467 at p=1.600e-01 sup/tau0= 3.9493 sup/max(tau0,d)= 3.9493 unimodal(inc then dec)=True  max 2Var=   7.3702 (log_d n)^2= 16.71
   elapsed 28s

== (4) K_n with the absolute gap: tau_abs(p) vs max(tau(p), (n-1)/(n-2)); n >= 3
n=3: max|tau_abs - max(tau, (n-1)/(n-2))| = 4.44e-16; tau_abs(0)=2.0000, tau_abs(1-)=2
n=4: max|tau_abs - max(tau, (n-1)/(n-2))| = 3.11e-15; tau_abs(0)=1.5000, tau_abs(1-)=3
n=5: max|tau_abs - max(tau, (n-1)/(n-2))| = 4.44e-15; tau_abs(0)=1.3333, tau_abs(1-)=4
n=8: max|tau_abs - max(tau, (n-1)/(n-2))| = 7.99e-15; tau_abs(0)=1.1667, tau_abs(1-)=7
n=16: max|tau_abs - max(tau, (n-1)/(n-2))| = 7.28e-14; tau_abs(0)=1.0714, tau_abs(1-)=15
exit=0
