1. Draw the lattice of subgroups (up to conjugacy) of .

The symmetry of a system of balancing balls is given by

are which can be thought of as the group

Now I know that, up to conjugacy, the subgroups of are , , , , , , , and .

However, I don't know how to construct a lattice of subgroups from this. How should it be structured?

2. For each of the subgroups write down, and justify, its Fixed Point Subspace when acting on the two dimensional surface given by .

What is a Fixed Point Subspace? How do you find it?

3. Draw the lattice of isotropy subgroups up to conjugacy.

Same problem as in 1.

If someone could enlighten me on this that would be great.

Thanks!