Summary table:
n.Systems...Tables....Group orders
1.......1........1....1
2.......1........2....1
3.......3.......10....1 2 6
4.......7.......92....1.2 2.3 6.2
5......22.....1321....1.5 2.10 4 6.4 20 24
6......77....27882....1.19 2.31 4.7 6.12 12.4 20 24.2 120
7.....314...819330....1.85 2.122 4.32 6.36 8.4 12.19 20.2 24.6 36.2 42 48 72 120.2 720
n is the size of the system.
Systems is the count of nonisomorphic systems of that size.
Tables is the total number of tables, with no culling for isomorphism.
Group orders is the number of systems with each size of automorphism group.
For example, there are 314 nonisomorphic Average systems with 7 elements.
85 of those systems have the trivial automorphism group (only the identity),
and each system gives rise to 7! = 5040 distinct tables. There's one
system with an automorphism group of 720 elements, which gives rise to only
5040/720 = 7 different tables. The total number of possible 7element tables
is 7^49, of which roughly 7^7 satisfy the Average rules.
We have the obvious identities 314 = 85 + 122 + 32 + ... + 1 + 2 + 1 and 819330 = 5040 * (85/1 + 122/2 + 32/4 + ... + 1/72 + 2/120 + 1/720).
