A Gallery of Groups · Hard problem

Realize the symmetries of a square as $2\times2$ matrices: $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ (a $90^\circ$ rotation) and $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ (a reflection), satisfying $R^4=F^2=I$. The $8$ elements of this group are $I,R,R^2,R^3,F,FR,FR^2,FR^3$ (indexed $0$ through $7$ in that order). Compute the matrix product $FR^3FR^2$ and give the index ($0$ to $7$) of the element it equals.