Table 1.

Two possible topological states for the symmetry indicator ${\mathbb{Z}}_{\mathrm{2,2,2,4}}=\left\{\mathrm{0,0,0,2}\right\}$ (47)

$\left({\nu }_0;{\nu }_1{\nu }_2{\nu }_3\right)$	$n_{2_{\left[1\overline{1}0\right]}}$	$n_{{\mathcal{M}}_{\left[1\overline{1}0\right]}}$	$n_{\mathcal{I}}$
(0;000)	1	0	1
(0;000)	0	2	1

$\left({\nu }_0;{\nu }_1{\nu }_2{\nu }_3\right)$ are the ${\mathbb{Z}}_2$ invariants for 3D ${\mathbb{Z}}_2$ TIs. $n_{2_{\left[1\overline{1}0\right]}}$ is a ${\mathbb{Z}}_2$ invariant for the twofold rotational symmetry $2_{\left[1\overline{1}0\right]}$. $n_{2_{\left[1\overline{1}0\right]}}=1$ corresponds to a rotational-symmetry–protected TCI, which features two unpinned Dirac surface states on the $\left(1\overline{1}0\right)$ surface. $n_{{\mathcal{M}}_{\left[1\overline{1}0\right]}}$ is the mirror Chern number for mirror plane ${\mathcal{M}}_{\left[1\overline{1}0\right]}$ and is a ${\mathbb{Z}}_N$ invariant. $n_{{\mathcal{M}}_{\left[1\overline{1}0\right]}}=N$ features N Dirac surface states. Finally, $n_{\mathcal{I}}$ is the ${\mathbb{Z}}_2$ classified inversion ($\mathcal{I}$) protected TCI index. $n_{\mathcal{I}}=1$ features the hinge state in a 3D finite geometry preserving $\mathcal{I}$.