If one were to lay out the dihedral group $D_4$ multiplication table, it might look like this (generators $a$ and $b$):
If we abstract from the generators and just call the elements $0,1,2,3,4,5,6,7$, we obtain a raw multiplication table for $D_4$ (which I do not display). My question is:
Q. Is there a canonical, preferred multiplication table using just integers to indicate the group elements for a finite group, or is the ordering of elements essentially arbitrary?
I am thinking of comparing multiplication tables of groups of the same finite order, and I am wondering if I can compare canonical tables, or should I instead compare all possible permutations of the tables?