= Solution
Act on the four labelled vertices. A proper rotation gives an even <permutation>, identifying the rotational <tetrahedral symmetry> group with $A_4$, an <alternating group>. Geometrically there are one identity, three half-turns about opposite-edge midpoint axes, and two nontrivial rotations about each of four vertex-face axes. Therefore
$$
\boxed{|G|=1+3+8=12.}
$$
The <tetrahedral rotation conjugacy classes> are
$$
\boxed{\begin{array}{c|l|c}
&\text{elements in vertex notation}&\text{size}\\\hline
C_0&\{e\}&1\\
C_2&\{(12)(34),(13)(24),(14)(23)\}&3\\
C_+&\{(123),(134),(142),(243)\}&4\\
C_-&\{(132),(143),(124),(234)\}&4
\end{array}}
$$
To justify the splitting, the <centralizer> of a 3-cycle within $A_4$ is its three-element <cyclic subgroup>, so its <conjugacy class> has size $12/3=4$. Its inverse is not conjugate to it in $A_4$: the permutations that reverse that cycle while preserving its fixed vertex are odd. Conjugating $(123)$ by the even permutations produces the displayed $C_+$, and taking inverses gives $C_-$. The three half-turns are conjugate, since rotations move their axes into one another. Their centralizer has order four and their class therefore has size three.
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