Why does the Gromoll-Meyer Sphere have dimension $7$?

Question

In exercise 10.33 of the book "Matrix Groups for Undergraduates" the Gromoll-Meyer sphere is described by taking the quotient of a smooth left action of $Sp(1) \times Sp(1)$ on the manifold $Sp(2)$ defined by $$(q_1, q_2) \circ A = \begin{pmatrix} q_1 & 0\\ 0 & q_1 \end{pmatrix} \cdot A \cdot \begin{pmatrix} \overline{q}_2 & 0\\ 0 & 1 \end{pmatrix}$$ where $\overline{q}$ denotes the conjugate of a quaternion $q$. The point of the exercise is to show that this action is free, but I am confused about the fact that the Gromoll-Meyer sphere is 7-dimensional. By the quotient manifold theorem, shouldn't the dimension of the Gromoll-Meyer sphere be $dim(Sp(2)) - dim(Sp(1) \times Sp(1)) = 10 - 6 = 4?$ I suppose I'm confused somewhere about the dimension of $Sp(1) \times Sp(1)$ or am misinterpreting the quotient manifold theorem.