Any information on Julia Fractal symmetry group?

What is the symmetry group of the julia set?

http://cs.clarku.edu/~djoyce/julia/julia2.gif

If you look at the picture there are a few obvious ones:

• order 2 rotate the whole thing around the center of any blob
• order 3 rotate around the meeting point of three blobs
• warping back and forth

basically what I was wondering is two things:

• how do you find the symmetry group
• how do you find an equation that warps the julia set back to itself according to some symmetry? (I want to make animations of that!)

but I am seriously stuck on this so basically any tips you can give me at all would be wonderful!