So I am going to be teaching the basics of group theory to high schoolers in a few weeks, and I want to hear what the Stack Exchange network has to say on the matter.

1. What are the applications and insights I should show them in order to excite them about the subject?
2. What would be a good possible outline of the syllabus, including the rote aspects of group theory, and what are some possible sample problems for the non-rote aspects, i.e. this is why group theory is nifty, these are the deeper connections, these are intriguing applications, etc.?

Thanks in advance!