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.