ag.algebraic geometry – Log canonical surface with an elliptic singularity

I would like to know if there is an example as follows:

$X$ is a log canonical surface and $x in X$ is an elliptic singularity such that

  1. The minimal resolution of $x$ is a circle of rational curves (or a single nodal rational curve).
  2. The singularity $x$ is non-$mathbb{Q}$-factorial.

I think it looks reasonable, but I do not know any explicit example of such a surface. I searched some papers but none of them discuss the $mathbb{Q}$-factoriality.