Embedding of finite group quotient of the projective space

Question

Let us consider the $\mathbb{Z}_2$ action on the complex projective space $\mathbb{P}^3$ defined by the involution $[Z_0, Z_1, Z_2, Z_3] \to [Z_1, Z_0, Z_3, Z_2]$. Let $Y$ be the quotient space.

Question: 1. What is the minimal $n$ such that $Y$ can be embedded in $\mathbb{P}^n$ ?

2. What is the embedding map ?

Answer 1

The minimal $n$ is 5. The map is given by the invariant quadrics.