Fibration with total space $SU(3)$, base $S^5$ and fiber $S^3$

Question

I would like to find a fibration with total space $SU(3)$, base $S^5$ and fiber $S^3$.

Any idea?

Answer 1

The action of $SU(3)$ on $\mathbb{C}^3$ preserves the unit sphere $S^5$, and the stabilizer of a point is $SU(2) = S^3$.