Let $S_n$ be the symmetric group of $n$ elements $\{ 1,2,...,n \}$ and $G$ be a subgroup of $S_n$ or order $n$. Denote the elements in $G$ by $\{ \sigma_1,...,\sigma_n \}$. 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 $A$ is symmetric, is the matrix $A$ invertible?