"Fundamental phenomena and laws of nature are related to symmetry and, accordingly, symmetry is one of science’s basic concepts. Perhaps it is so important in human creations because it is omnipresent in the natural world." This quote is from the introduction of the book "Symmetry through the Eyes of a Chemist." It is a wonderful book with great illustrations from art, architecture, nature etc.
As mentioned above, symmetry is everywhere. You can find it in math, biology, biochemistry and of course chemistry. The good thing about it is once you learn it you will never forget it and you will see how useful it is.
There is an excellent website here: http://symmetry.otterbein.edu/
It's a great source for point group practice and learning. Technische Universitat Darmstadt also has some good information and tutorials about symmetry and point group theory (it is in English) : Here
Let me continue with the post now. In general, if a molecule;
1. has two or more C5 axes, the point group is : Ih
2. has two or more C4 axes, the point group is : Oh
3. has two or more C3 axes, the point group is : Td
Now, how do we determine the point group of a molecule (or an object)?
I will write down the way I learned it and it works perfect.
First, find the principal axis. This is the axis with the highest order of axis. Suppose that the molecule has C2, C3 and C4 axes. So, C4 is the principal axis.
Second step: Check if there is a perpendicular C2 axis to the principal axis. If there is one (only one is enough), the point group is Dn. If you can't find any, then the point group is Cn.
Third step: Now it is time to check for a plane of symmetry. We are looking for the σh (Sigma h) now. This is a mirror plane perpendicular to the principal axis. If you find it, then the point group is either Cnh or Dnh.
If a perpendicular plane of symmetry is missing, we look for a mirror plane that is not perpendicular to the principal axis. If there is one then it is either Cnv or Dnd (don't ask why it is so. That's how it is called.) If you can't find another mirror plane, then the point group is Cn or Dn.
It's actually very simple to determine the point group of a molecule. But, you have to be very careful. Some molecules are really tricky. Let's do an example I am just making up a molecule for practice. L and X are different ligands.