Let $S_n$ be the symmetric group on $n$ elements $\{ 1,2,\dotsc,n \}$ and $G$ be a subgroup of $S_n$ of order $n$. Denote the elements in $G$ by $\{ \sigma_1,\dotsc,\sigma_n \}$. Let the matrix $A=(\sigma_i(j))\in M_{n\times n}(\mathbb Q)$. In general, $A$ is not invertible as pointed out by Dave.

If $G$ is a transitive subgroup of order $n$, is the matrix $A$ invertible?