We define the non-exterior square graph $\hat{\Gamma}_G$ which is a graph associated to a non-cyclic finite group with the vertex set $G\backslash\hat{Z}(G)$, where $\hat{Z}(G)$ denotes the exterior centre of $G$ and two vertices $x$ and $y$ are joined whenever $x\wedge y\neq1$ wheren $\wedge$ denotes the operator of non-abelian exterior square. In this paper, we investigate how the group structure can be affected by the planarity, completeness and regularity of this graph.