In line geometry, oriented line transformations can be induced by a pin group enlarging \(\textit{Pin}(3,3)\) by introducing a group anti-automorphism, called symplectic pin group \(\textit{Pin}^{sp}(3,3)\) , which double covers the group of orthogonal and anti-orthogonal transformations of \({\mathbb {R}}^{3,3}\) . This paper explores the topology of these groups, in particular the connectivity structure and Pauli split structure over the connected component containing the identity. In quantum computation, constructing geometric models for 2-qubit systems is important not only for understanding the unitary evolutions, but also for efficient algorithm design. This paper presents a construction of pin group over \(\textit{Pin}(6)\) in a 3-qubit system, which has the same connectivity structure and Pauli split structure as \(\textit{Pin}^{sp}(3,3)\) , and which may help constructing projective geometric model of 2-qubit dynamics.