# Sharp bounds for the number of groups of order $75600$?

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 ?

• 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. – ahulpke Apr 19 '16 at 20:13
• 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). – Alexander Konovalov Apr 20 '16 at 22:19