As Anweshi noted a moment ago, a classic answer is the use of character tables by chemists (as explained in http://www.amazon.com/Group-Theory-Chemistry-David-Bishop/dp/0486673553/ref=pd_sim_b_2>this this book, for instance). The symmetry group of a molecule controls its vibrational spectrum, as observed by IR spectrosocopy. When Kroto et al. discovered $C_{60}$, they used this method to demonstrate its icosahedral symmetry.

As Anweshi noted a moment ago, a classic answer is the use of character tables by chemists (as explained in http://www.amazon.com/Group-Theory-Chemistry-David-Bishop/dp/0486673553/ref=pd_sim_b_2>this book, for instance). The symmetry group of a molecule controls its vibrational spectrum, as observed by IR spectrosocopy. When Kroto et al. discovered $C_{60}$, they used this method to demonstrate its icosahedral symmetry.