Consider a square. Split every edge in half. Then lable each half-edge clockwise using the elements of the following $Y = [1, 2, 3, 4, 5, 6, 7, 8]$ which denote the set $Y$. Then the elements of $Y$ are permuted by the elements of the symmetry group (dihedral group) $D_8$.
In cycle notation write the 8 permutations of $Y$.
Write the effect of 8 elements of D8 to Y in cycle notation.
Thank you to Bungo for the great guideline:
90 degree clockwise is:
(1 2 3 4 5 6 7 8) (3 4 5 6 7 8 1 2) Which returns us with (1 3 5 7) (2 4 6 8)
90 degree anticlockwise is: (1753)(2864)
Question now solved. Special thanks to Bungo for guiding me on the right path.
Start by drawing a picture:
We want to know what happens to the segments $1,2,3,4,5,6,7,8$ when $D_8$ acts in the usual way on the square.
Let's consider rotation clockwise by 90 degrees. This will move $1$ to $3$, and $3$ to $5$, and $5$ to $7$, and $7$ to $1$. Similarly, it will move $2\mapsto 4 \mapsto 6 \mapsto 8$. Therefore in cycle notation, this rotation is $(1357)(2468)$.
Next consider rotation by 180 degrees. This will move $1$ to $5$, and $5$ to $1$. Similarly, it will exchange $2$ and $6$, and it will exchange $3$ and $7$, and it will exchange $4$ and $8$. So, in cycle notation, rotation by 180 degrees is $(15)(26)(37)(48)$.
The other six elements of $D_8$ are the identity, rotation counterclockwise by 90 degrees, and flips across the horizontal, vertical, and the two diagonal axes. See if you can do these for yourself, and I encourage you to update your question if you get stuck or want to check your answers.