respectively; for two transverse orders, eigenstates of of $T_{111}$, $T_{\overline{1}11}$, $T_{1\overline{1}1}$, and $T_{11\overline{1}}$, are assigned to sublattices 1, 2, 3 and 4, and to 1, 4, 2 and 3, respectively.

Introducing the quadrupole operators, $$Q_{p} = \left\{ \begin{array}{lcl}
\frac{1}{\sqrt{3}} \left( Q_{yz}+Q_{zx}+Q_{xy} \right) & {\textrm for} & 
     p=111, \\
\frac{1}{\sqrt{3}} \left( Q_{yz}-Q_{zx}-Q_{xy} \right) & {\textrm for} & 
     p=\overline{1}11, \\
\frac{1}{\sqrt{3}} \left(-Q_{yz}+Q_{zx}-Q_{xy} \right) & {\textrm for} & 
     p=1\overline{1}1, \\
\frac{1}{\sqrt{3}} \left(-Q_{yz}-Q_{zx}+Q_{xy} \right) & {\textrm for} & 
     p=11\overline{1}, \\
\end{array} \right.$$ we can construct the quadrupole ordering state by assigning them to each sublattice in the same way as for octupole orderings. Since $Q_p$’s and $T_p$’s are simultaneously diagonalized because of commuting with each other, $Q_p$ could be represented as $$Q_p = 
\left(\begin{array}{cccc}
    -q_1 & 0 & 0 & 0 \\
    0 & -q_1 & 0 & 0 \\
    0 & 0 & +q_1 & 0 \\
    0 & 0 & 0 & +q_1 \\
\end{array} \right),$$ with $q_1=-8.273$.

Let the octupole ordering be primarily realized. Then, each Np ion is in the eigenstate of the eigenvalue $-t_1$ (or $t_1$). Since the state is also the eigenstate of the eigenvalue $-q_1$, the quadrupole ordering is simultaneously induced. On the other hand, if the 