Truncation of $D=5$, ${cal N}=8$ Supergravity by $mathbb Z_2^3$

The scalar manifold of $D=5, mathcal N=8$ SUGRA is

$$mathcal M = frac{E_{6(6)}}{Usp(8)}$$

where $USp(8)$ is a maximal compact subgroup of $E_{6(6)}$
and the 42 scalars of the theory correspond to the noncompact generators of $E_{6(6)}$ orthogonal to those of $USp(8)$.

In this paper (Appendix B), the authors described a truncation of this scalar manifold from the above to the following manifold

$$mathcal M’ = [SO(1,1)times SO(1,1)] times frac{SO(4,4)}{SO(4)times SO(4)}$$

by keeping only those scalars invariant under a $mathbb Z_2^3 subset SO(6)times SL(2)$. These three $mathbb Z_2$ are generated by the following generators $P_1 Q, P_2 Q, P_3 Q$ where:

$$P_1 = text{diag}{-1,-1,1,1,1,1}subset SO(6)$$

$$P_2 = text{diag}{1,1,-1,-1,1,1}subset SO(6)$$

$$P_3 = text{diag}{1,1,1,1,-1,-1}subset SO(6)$$

$$Q = text{diag}{-1,-1} subset SL(2)$$

So my question is: Is there a way to derive, group-theoretically, the resulting scalar manifold, $mathcal M’$, from the action of projecting out $$mathbb Z_2^3subset SO(6)times SL(2)subset E_{6(6)}~?$$