Group theory is a branch of mathematics which studies symmetry. Here one studies symmetries of objects and interrelationships among symmetries. Now rotate it through 90 degrees anticlockwise and then flip it along the length of Monticello. Thus the two original symmetries are related to this third one in the manner we saw.
Representation theory is a powerful way of studying groups using linear algebra.
The idea is somewhat similar to representing a very complicated physical object by taking its pictures from various angles. The pictures may make it easier to understand the spatial interrelationships among various parts of that object. Lie groups and Algebraic groups are very special types of groups. They are technical notions encapsulating some of the most important kinds of symmetries of space which occur in real life.