be revealed by the representation of charge operator $Q_{p}=$ diag$\left( 1,\omega
_{p},\omega _{p}^{2},\cdots ,\omega _{p}^{p-1}\right) $ as $$\left( Q_{p}\otimes \mathbbm{1}\right) ^{-1}A^{[i]}\left( Q_{p}\otimes \mathbbm{1}\right) =\omega _{p}^{|i|}A^{[i]}.$$For the later discussion, it is useful to introduce another $p\times p$ matrix $$Y_{p}=\left[ 
\begin{array}{ccccc}
0 & 1 & 0 & \cdots  & 0 \\ 
0 & 0 & 1 & \cdots  & 0 \\ 
0 & 0 & 0 & \cdots  & 0 \\ 
\vdots  & \vdots  & \vdots  & \ddots  & 1 \\ 
1 & 0 & 0 & 0 & 0\end{array}\right] ,  \label{Y}$$as the regular representation of generator of $\mathbb{Z}_{p}$ symmetry. It will be frequently used and played a significant role in the following classification.

[|C[2cm]{}|C[0.8cm]{}|C[0.8cm]{}|C[0.8cm]{}|C[0.8cm]{}|C[0.8cm]{}|C[0.8cm]{}|]{} & 0 & $p-1$& $p-2$ &$p-3$ &$\cdots$& 1\
0 & 0 & $p-1$& $p-2$ &$p-3$ &$\cdots$&1\
1 & 1 & 0 & $p-1$&$p-2$ &$\cdots$ &2\
2 & 2 & 1 & 0&$p-1$ & $\cdots$&3\
3 & 3 & 2 & 1&$0$ & $\cdots$&4\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$&$\vdots$&$\ddots$&\
$p-1$ & $p-1$ & $p-2$ & $p-3$&$p-4$&$\cdots$&0\

So the parafermionic MPS can be constructed by taking graded tensor product of local tensors and contracting the virtual bonds between nearby local tensors with the homomorphism $\mathcal{C}$. The contraction 