Controlled not connectivity in the Clifford group

The Clifford group is the set of gates generated by CZ gates and the two local gates P $=$ \textbraceleft \textbraceleft 1,0\textbraceright ,\textbraceleft 0,i\textbraceright \textbraceright and H, the 2 by 2 Hadamard gate. It is known that, for a two qubit system, the Clifford group C$_{\mathrm{2}}$ is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC$_{\mathrm{2}}$ is a subgroup of order 4608 of the group C$_{\mathrm{2}}$. In order to better understand the set C$_{\mathrm{2}}$, we make two matrices U$_{\mathrm{1}}$ and U$_{\mathrm{2}}$ in C$_{\mathrm{2}}$ equivalent if U$_{\mathrm{1}} \quad =$ V U$_{\mathrm{2}}$ for some V $\in $ LC$_{\mathrm{2}}$. We show that this equivalence relation splits C$_{\mathrm{2}}$ into 20 orbits, O$_{\mathrm{1}}$, . . . ,O$_{\mathrm{20}}$, each with 4608 elements. Moreover, for each orbit O$_{\mathrm{i}}$, CZO$_{\mathrm{i}}$ intersects 9 different orbits O$_{\mathrm{i1}}$, . . . ,O$_{\mathrm{i9}}$ where O$_{\mathrm{ij}}$ does not equal O$_{\mathrm{i}}$ and with CZO$_{\mathrm{i}} \quad \cap $ O$_{\mathrm{ij}}$ containing 512 matrices for each j $=$ 1, 2, . . . , 9. The link \underline {https://www.youtube.com/watch?v}$=$\underline {lcYtB2tnXFw} leads you to a YouTube video that explains the most important results in this paper.