I have [asked this question on Math.StackExchange](https://math.stackexchange.com/questions/2729335/is-the-quotient-of-a-toric-variety-by-a-finite-group-still-toric), but haven't got any reply.


Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $\phi$ is a representation of finite group $G$ in $N \otimes\_{\mathbb{Z}} \mathbb{R}$,
\begin{equation}
\phi:G \rightarrow \text{GL}( N \otimes\_{\mathbb{Z}} \mathbb{R})
\end{equation}
which sends the lattice $N$ to $N$, and the cones of $\Sigma$ to cones of $\Sigma$, therefore each $\phi(g)$ defines a toric automorphism of $X$.


**Question 1**: Is the quotient variety $X/G$ (which exists from GIT) still a toric variety? Is the quotient morphism $X \rightarrow X/G$ toric?


**Question 2**: If so, how to construct the fan of $X/G$ and the toric morphism $X \rightarrow X/G$?


Any references on the two questions?



Is the quotient of a toric variety by a finite group still toric
No. If $X$ is the 47-dimensional torus and $G=\mathbb{Z}/47\mathbb{Z}$ acting by the permutation representation, then $X/G$ is not a rational variety (R. Swan, Inv. math. 7, 148-158 (1969)), therefore not toric.


