I would like help with the following problem:
Let $G$ be the subgroup of $GL_3(\mathbb{C})$ generated by the three matrices $ A=\begin{pmatrix} 0&0&1\\0&1&0\\1&0&0\end{pmatrix}\;,\quad B=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}\;,\quad C=\begin{pmatrix} i&0&0\\0&1&0\\0&0&1\end{pmatrix}$ 1. Compute the order of $G$. 2. Find a matrix $G$ of largest possible order (as an element of $G$) and compute this order. 3. Compute the number of elements in $G$ with this largest order.
So I've found the relations $A^2=B^3=C^4=I$, so I know my group so far consists of $\{I,A,B,B^2,C,C^2,C^3\}$, but I don't know how to proceed from here. I know the group is not abelian, but I just thought I'd try listing all two element products, so I have 30=6(5) choices for that (assuming, which I haven't checked, they are all distinct). I am just wondering if there is a more concrete way to approach this problem. I'd also like help with the other parts afterward. I can see that $BC$ and $CB$ have order 12, but I'm not sure what to do next.