In the multiplicative group $F^{\times}$of a field $F$ of order 43, let $\alpha$ have order 7 and let $\epsilon$ have order 3 . Working in the group $G L(3,43)$, let
$$
a=\left[\begin{array}{ccc}
\alpha & 0 & 0 \\
0 & \alpha^4 & 0 \\
0 & 0 & \alpha^2
\end{array}\right] \quad \text { and } \quad b=\left[\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
e & 0 & 0
\end{array}\right] \text {. }
$$
Show that $A=\langle a, b\rangle$ is noncyclic of order 63 , and that its natural action on the vector space $V$ of order $43^3$ is Frobenius.