= Solution
First, the <orbit-stabilizer theorem> applied to a face gives $|G|=20\cdot3=60$: its <stabilizer subgroup> consists of the identity and the two <rotations> through $120^\circ$ about the axis through the centres of that face and its opposite. A <rotation> preserving a face must preserve its outward normal and cyclically permute its three vertices, so there are no further stabilizing <rotations>.
The <conjugacy classes> are as follows:
* The identity: \b[size $1$].
* Half-turns about axes through midpoints of opposite edges: \b[size $15$].
* <Rotations> through $\pm2\pi/3$ about opposite-face axes: \b[size $20$].
* <Rotations> through $\pm2\pi/5$ about opposite-vertex axes: \b[size $12$].
* <Rotations> through $\pm4\pi/5$ about opposite-vertex axes: \b[size $12$].
There are $30/2=15$ edge axes, $20/2=10$ face axes, and $12/2=6$ vertex axes. They give respectively $15$, $2\cdot10=20$, and $4\cdot6=24$ distinct nonidentity <rotations>, already accounting for all $59$ nonidentity elements.
To justify the class assertions, conjugation by a <rotation> carries the <rotation> axis to its image and preserves the <rotation> angle about the transported oriented axis. The <transitive group action> on edges puts all half-turns in one <conjugacy class>. The <transitive group action> on faces puts the $120^\circ$ <rotations> defined using outward face normals in one <conjugacy class>; opposite faces encode opposite signs about the same unoriented axis. The same reasoning with outward vertex directions gives a class of twelve $72^\circ$ <rotations> and a class of twelve $144^\circ$ <rotations>, including their inverses. These last classes cannot merge: the <matrix trace> of a three-dimensional <rotation> is $1+2\cos\theta$, which differs for $\theta=2\pi/5$ and $4\pi/5$. Thus the list is complete.
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