How can I get sharp bounds for $gnu(75600)$, the number of groups of order $75600$.

I tried to determine the number of groups of order $15120$ to get a reasonable lower bound, but I quit GAP after some hours, noticing that there still was a long way to finish the calculation.

I determined $gnu(2160)=3429$ , $gnu(3024)=4635$ and $gnu(5040)=4539$, so a lower bound of $gnu(75600)$ is $gnu(3024)\cdot gnu(25)=9270$.

Can anyone give better bounds, or even the actual value ?


There are (usual method -- calculation with GAP) 22758 groups of order 15120. The calculation took about 2 days. I don't see a fundamental obstacle to running order 75600, but that might take a week or two.

  • $\begingroup$ Did you check that the groups are actually non-isomorphic ? $\endgroup$ – Peter Apr 19 '16 at 15:57
  • $\begingroup$ This is the result of ConstructAllGroups which already uses some isomorphism tests, I did not try to determine distinguishing invariants or ran explicit isomorphism tests on the result. $\endgroup$ – ahulpke Apr 19 '16 at 20:13
  • $\begingroup$ For me, r:=ConstructAllGroups(15120) took 72hrs, the resulting list has length 22758 and ForAll(r,IsGroup) returns true meaning that the result has no lists of groups that the algorithm did not not manage to distinguish (otherwise, the list r would have elements which are lists themselves). $\endgroup$ – Alexander Konovalov Apr 20 '16 at 22:19
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    $\begingroup$ @ahulpke: I've added an entry for gnu(15120) at github.com/alex-konovalov/gnu/blob/master/gnudata.g. Thinking of crowdsourcing the database of gnu(n). $\endgroup$ – Alexander Konovalov Apr 20 '16 at 22:30
  • $\begingroup$ @AlexanderKonovalov Thank you for cross-checking the result. It would be very nice if a database of the known gnu's would be available. $\endgroup$ – Peter Apr 21 '16 at 16:34

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