Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication table of the elements from this.
One then goes and looks up this group from some kind of database or such and gets the multiplication table of the elements of the same group from there.
However, comparing the two tables, one notices that they look completely different. The reason for this is simple, the elements are ordered differently, thus the results of the multiplication are different.
Given a group, does there exist a standard or canonical ordering of the elements? If I have some ordering of the elements $\omega$, is there a simple way to, using another ordering $\Omega$, permutate $\omega$ to get $\Omega$ and thus get the same multiplication table?
PS. Is this kind of changing the order ever important? The groups are the same, in the end.