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Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the permutation representation of this action, to show that T is isomorphic to a subgroup of $S_4$.

Outline of my attempt at a proof:

Let $S$ be the set of faces of the tetrahedron. There exists a homomorphism $f:T$ $\rightarrow$ (permutations of the faces). There are $4$ faces, so $|Perm(|4|)| = 24$. We want to show that $T$ is a bijection, and show the faithfulness of $f$. Well, $|Ker(f)|=1$, because no rotation fixes all faces. So $f$ is injective. Also, the class equation for $T$ is $12 = 1+3+4+4$, which admits no "subsums" to $6$, implying that there are no odd permutations because $T$ has no subgroup of index $2$. So $T \approx A_4$, which is a subgroup of $S_4$.

The thing that evades me is how to prove that $f$ is surjective?