Cube/Rotation at space diagonal and axis through edge midpoints/Induced permutation/Exercise

We consider the cube.

Let ${}\alpha$ be the rotation of the cube around the axis given by the vertices ${}A$ and ${}G$ that sends the vertex ${}B$ to ${}D$. Let ${}\beta$ be the half rotation around the vertical axis (that is, the line connecting the center of the face ${}A,B,C,D$ and the center of the face ${}E,F,G,H$).

a) Establish the value tables for the permutations on the set of vertices ${}\{A,B,C,D,E,F,G,H\}$ induced by ${}\alpha ,\beta ,\alpha \beta$, and ${}\beta \alpha$.

b) Determine the rotation axis of ${}\alpha \beta$, and of ${}\beta \alpha$. Determine also the orders of these rotations.

c) Determine the cycle representation of the permutation on the set of vertices induced by ${}\alpha ^{2}$. What is ${}\alpha ^{1001}$?

d) We consider the permutation ${}\sigma$ that is given by the value table

${}x$	${}A$	${}B$	${}C$	${}D$	${}E$	${}F$	${}G$	${}H$
${}\sigma (x)$	${}B$	${}C$	${}D$	${}A$	${}G$	${}H$	${}E$	${}F$

Does there exist a rotation of the cube that induces this permutation? Compute the sign of ${}\sigma$.

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