What is the precise definition of a higher symmetry in group theory? I see this term scattered around, such as in Cotton's Chemical Applications of Group Theory:
...In the cyclic groups each of the operations, Cn, Cn2, Cn3, ..., , Cnn-1, constitutes a class by itself and we continue to use this notations. However, in all other groups of higher symmetry, the number of classes spanned by these operations will be reduced in the following way...
Does higher symmetry strictly mean a supergroup? Or is it a quantitative statement about something like the number of symmetry elements or number of irreducible representations (which are not always ordered the same way, as in the case of C3v and S4).
I am failing to find intuition because, if it does mean strictly supergroups, we can't compare the "amount" of symmetry in C2 compared to C3 (or Td compared to Ci even!).