Let $S_{(2)}^{2}$ denote a sphere of unit radius, centered at the origin of threedimensional space, with two

Question:

Let $S_{(2)}^{2}$ denote a sphere of unit radius, centered at the origin of threedimensional space, with two noncoincident points on its surface. What is the group of transformations that leave $S_{(2)}^{2}$ invariant? Does the group depend on the relative position of the two points? Does it depend on whether the two points are distinguishable or indistinguishable?

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