Let $f$ and $g$ be complex-valued functions on a finite group $G$. Left convolution by $f$ can be realized as an operator $L_f(g) = f * g$, so it follows that $L_f$ can be represented as a matrix ($f$ and $g$ can be viewed as vectors in $\mathbb{C}^n$ by listing the group $G = \{x_1, x_2, ... x_n\}$ and setting the ith component of the vector $f$ as $f_i = f(x_i)$). Is there any easy way of writing down this matrix? From looking online it seems that convolution with respect to $\mathbb{Z}$ between finite sequences can be represented by a Toeplitz matrix, and I'm curious if this is still true when extending to functions on arbitrary finite groups.
The reason I'm hoping that this is possible to do with Toeplitz matrices is because Sage has nice pre-built functions for constructing such matrices, and this is for a project being written in Sage.