Writing every element in a group from its presentation

Question

Problem

If we have a group $G$ of order $8$ such that $G=\langle i,j,k | ij=k,jk=i,ki=j,i^2=j^2=k^2\rangle $, and we denote $i^2=m$, show that every element of G can be written in the form $e,i,j,k,m,mi,mj,mk$. The problem suggests writing out the multiplication table during the course of the solution.

Solution attempt with questions

(I will edit with the multiplication table added once I convert it from $\LaTeX$ to markdown.

First question: How does this group have order 8? I can only find 7 elements: $e,i,j,k,i^2,j^2,k^2$. It doesn't make sense for the 8th element to be $m$ as $m$ is just a denotion for $i^2$.

Next question:

I've shown all the trivial answers:

Given by presentation:

$ij=k,jk=i,ki=j, ii=m, jj=m, kk=m$

Extrapolated with $i^2=j^2=k^2=m$:

$i^2i=mi$, $k^2i=mi$, $j^2i=mi$ (I use similar logic to find terms like $i^2j$ and $i^2k$)

Elements of form $x^3$:

$i^2i=(ii)i=mi, j^2j=mj, k^2k=mk$

Where I'm stuck at is determining what the elements like $i^2i^2$ are. For example, $i^2i^2=iiii$. There isn't a part of the presentation that helps reduce $i^2$ to a usable form.

If I'm looking at this completely wrong, I'd appreciate some advice or a hint towards the solution.

Answer 1

1. Since $i^2=j^2=k^2=m$, then $mi=im$, $jm=mj$, $km=mk$.
2. Since $m=kk=kij=(ij)(jk)(ki)=im^2i=m^3$, then $m^2=e$.
3. Since $i^4=e$, then $i^{-1}=i^3=im$. Similarly, $j^{-1}=jm$, $k^{-1}=km$. So, $ji=ijm$, $kj=jkm$, and $ik=kim$.
4. Using 1-3 we can simplify any word from $i,j,k,i^{-1},j^{-1},k^{-1}$. Hence, here is the complete list of elements of group $G$: $e,i,j,k,m,mi,mj,mk$.