Todd-Coxeter Algorithm for D4 I am getting stuck on using the Todd-Coxeter algorithm for the group D4. Any help would be appreciated.
Problem statement: Use T-C to analyze the group G generated by x and y with relations $x^4, y^2, yxy^{-1}x$ as follows. Write out tables for the action of G on the cosets of $\langle x \rangle$.
My work: Let $G$ be the group $\langle x,y|x^4,y^2,yxy^{-1}x \rangle$ and let H=$\langle x \rangle$.
Tables: x x x x           y y                 y x y^(-1) x
       1 1 1 1 1         1 2 1               1 2 2      1 1
       2 2 2 2 2         2 1 2               2 1 1      2 2

So I am getting that $x=(1), y=(1 2)$. Since there are two indices, the index of H is 2. However, x is supposed to have order 4 in order for D4 to have order 8.
But x doesn't seem to have order 4? So have I done something wrong? 

You certainly have got this table besides the table above:

Let $x=h$ so $1h=h\cdot1$ and then $1x=h\cdot1$. Also $2y=1$ and $2x=1x^{-1}y^{-1}=h^{-1}\cdot2$. These results make the latter table to be like:

According to table in the body above, we see that $1xxxx=1$. The second table makes it to $$h^4\cdot1=1\longrightarrow h^4=1$$ Also $$2xxxx=2\Longrightarrow h^4\cdot 2=2\longrightarrow h^4=1$$ I think these results may help us to be sure of $H$'s presentation. However, as @Derek commented, this method is silent about the order of $H$.
