1)
In general $|s| = 2$ by (2). And we can compute
- For $D_6$ we have the following elements
$$
\{ 1,r,r^2 , s , sr , sr^2 \}
$$
Now $|1| = 1, |r| = 3$. And we have that
$$
(r^2)^2 = r^4 = r
$$
So $|r^2| = 3$. $s(rs)r = ssr^{-1}r = 1$, so $|sr| = 2$. We have that
$$
s (r^2 s) r^2 = ss r^{-2}r^2 = 1
$$
So $|sr^2| = 2$. In total
- For $D_8$ we have the following elements
$$
\{ 1,r,r^2,r^3,s,sr,sr^2,sr^3 \}
$$
here $|r| = 4$. We have that $(r^2)^2 = 1$. So $|r^2| = 2$. We have that $|r^3| = 4$. We have that every element on the form $sr^i$ has order $2$. That is
$$
s(r^i s) r^i = ss r^{-i} r^{i} = 1
$$
So in total
1 | r | r^{2} | r^{3} | s | sr | sr^{2} | sr^{3} |
1 | 4 | 2 | 4 | 2 | 2 | 2 | 2 |
- For $D_{10}$ we have the following elements
$$
\{ 1,r,r^2,r^3,r^4,s,sr,sr^2,sr^3,sr^4 \}
$$
We have $|r| = 5$. We have that $|r^2| = 5$. We have that $|r^3| = 5$. And $|r^4| = 5$. In total we get
1 | r | r^{2} | r^{3} | r^{4} | s | sr | sr^{2} | sr^{3} | sr^{4} |
1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 |