So I have this question to solve. I've already shown that the group of rotations of a cube is isomorphic to $S_4$. I need to prove that these two groups are not conjugate when considered as subgroups of the group of isometries of 3D space. First off, I'm having a hard time figuring out how $S_4$ can be a subgroup of the group of isometries of 3D space. All the isometries in 3D can be represented as a product of reflections, so, this group would be a group containing the reflections in 3D. Obviously, I can picture this as $S_4$ representing the symmetries of a cube, but I don't see how that would help me solve the question. Could anybody give me some tips as to how to approach this question?
2026-03-28 05:23:17.1774675397
How is a symmetric group the subgroup of the group of isometries of three-dimensional space?
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Presumably the question is referring to the other faithful representation of $S_4$ in three dimensions, namely the group of all symmetries of a regular tetrahedron.