fermions have proper charge conjugation symmetry.

               $N_f$                                       $\mathcal{S}$$\&$$\mathcal{I}$-subgroup                                                    $\mathcal{R}$-subgroup                  $P$          $C$        $SW_{4}$   
  ----------- ------- ------------------------------------------------------------------------------------------------------------------ ------------------------------------------------ ------------ ------------ ------------ --
   Staggered    $4$                                        $\mathcal{S}_{\mu}$,$\mathcal{I}_{\mu}$                                                    $\mathcal{R}_{\mu\nu}$               $\bigcirc$   $\bigcirc$   $\bigcirc$  
     Adams      $2$                       $\mathcal{S}_{\mu}\mathcal{S}_{\nu}$, $\mathcal{S}_{\mu}\mathcal{I}_{\mu}$                                  $\mathcal{R}_{\mu\nu}$               $\bigcirc$   $\bigcirc$   $\bigcirc$  
   Hoelbling    $1$    $\mathcal{S}_{\mu}\mathcal{S}_{\nu}\mathcal{S}_{\rho}\mathcal{S}_{\sigma}$, $\mathcal{S}_{\mu}\mathcal{I}_{\mu}$   $\mathcal{R}_{\mu\nu}\mathcal{R}_{\rho\sigma}$   $\bigcirc$   $\bigcirc$    $\times$   

We lastly consider hypercubic symmetry. In staggered fermions, the rotation Eq. (\[rot1\]) and axis reversal Eq. (\[axis1\]) form hypercubic symmetry [@KilSha], which enhances to Euclidian Lorentz symmetry in the continuum limit. In the case of the Adams-type fermion, the action is invariant under the rotation Eq. (\[rot1\]) and the shifted-axis reversal $\mathcal{S}_{\mu}\mathcal{I}_{\mu}$. These two symmetries can form proper hypercubic symmetry $SW_{4}$ in this case. Thus we conclude that Adams fermion recovers Lorentz symmetry in the continuum. In the Hoelbling-type formulation, the action breaks the rotation symmetry into its subgroup called doubled rotation [@Hoel], which is given by $$\mathcal{R}_{\rho\sigma}\mathcal{R}_{\mu\nu}\sim
\exp[{\pi\over{4}}(\Gamma_{\rho}\Gamma_{\sigma}+\Gamma_{\mu}\Gamma_{\nu})]
\exp[{\pi\over{4}}(\Xi_{\rho}\Xi_{\sigma}+\Xi_{\mu}\Xi_{\nu})]
\phi(R_{\rho\sigma}^{-1}R_{\mu\nu}^{-1}p)
\ ,
\label{DR}$$ where $(\mu,\nu,\sigma,\rho)$ is any permutation of ($1,2,3,4$). It is also invariant under sequential transformations of ($\mu,\nu$ rotation), ($\nu,\mu$ rotation), ($\mu$ shift) and ($\nu$ shift) as $$\mathcal{S}_{\nu}\mathcal{S}_{\mu}\mathcal{R}_{\nu\mu}\mathcal{R}_{\mu\nu}\sim
\exp(ip_{\mu}+ip_{\nu})\Gamma_{\mu}\Gamma_{\nu}\,\phi(\tilde{p})
\ ,
\label{SDR}$$ with $\tilde{p}_{\mu,\nu}=-p_{\mu,\nu}, \tilde{p}_{\tau}=p_{\tau}$, $\tau\not=\mu,\nu$. The loss of rotation symmetry indicates that we cannot define hypercubic symmetry in the Hoelbling fermion. It implies that 