My friend and I think of a smooth analog of Rubik's cube. An idea is:
Consider the 2dimensional sphere $ S ^ {2} $. We choose three parameters: $ (L, H, theta) $. Right here $ L $ is a ray that crosses the origin, $ H $ is a plan that crosses $ L $, the sphere and orthogonal to $ L $. $ theta in [0, 2pi]$ is an angle that we will rotate the part of the sphere. Here is the photo for action:
Now, I have some questions about a group $ G $ generated by this action on $ S ^ {2} $.

is $ G $ has a finite dimensional Lie group? If so, what is the size of the group?

Is there an interesting and nontrivial relationship between the elements? For example, in $ mathrm {SO} (3) $the composition of two rotations is also a rotation. But our group clearly does not satisfy such things.
edit. As Elkies said, it is an infinite dimensional group, because it contains an infinite dimensional abelian group (generated by elements to $ L $ and variant $ H & theta $. But we can think of certain subgroups that seem to be finite or even finite.
Before doing this, there is a minor problem with the definition of $ G $ that Jim Conant mentioned. I want to ignore the action that moves only measure 0 parts, like an equator.
To do this, for any two functions $ g_ {1}, g_ {2}: S ^ {2} to S ^ {2} $ we can define an equivalence relation as
$$
g_ {1} sim g_ {2} text {iff} {x in S ^ {2} ,: , g_ {1} (x) neq g {2} (x) } text {has a measure 0}
$$
Now consider the group quotient by this equivalence relation, we will then have a more reasonable group. There is a reason why I think this is reasonable – there are certain subgroups I want to think about.

Suppose we allow only "halfspheres". We therefore consider the subgroup of $ G $ where the plane $ H $ should pass the origin.
Is it a finite dimensional group? I think $ mathrm {SO} (3) $ is a codimension 1 subgroup of this halfsphere group, but I'm not sure.

What will be the finite subgroups of this $ G $? There is a natural but not trivial way to build a finite subgroup of $ G $: think real Riddles of the Rubik cube type. For example, place the Rubik's cube in the center of the sphere and bend it along the planes that correspond to the sliced sides of the Rubik's cube (I hope you understand what I mean), then turn them around. pieces along the axis of the cube. In other words, it is an isomorphic subgroup to the group of Rubik's cube. We can do the same with Pyraminx, Metaminx, Dogic, Skewb or any ordinary polytope. I want to know if they cover all possible finite subgroups of $ G $.

What about higherdimensional spheres, or other highly symmetrical varieties (like a torus or hyperbolic spaces)?