  A plaquette-centered rotation symmetry $C_{4}^{p}$ is identified to play a
significant role in determining and stabilizing the Fermi-surface structure of
Fe-based superconductors. Together with the $S_{4}$ symmetry previously found,
we are able to sort out the tangling orbitals and solve the puzzle of pairing
symmetry of superconductivity in $\text{KFe}_{2}\text{As}_{2}$ in a simple but
comprehensive way. By modeling the material with a strong coupling
$t-J_{1}-J_{2}$ model, we find phase transitions of pairing symmetry driven by
the competition between the local spin antiferromagnetic couplings from nodal
$d_{x^2-y^2}\times s_{x^2+y^2}$-wave to nodeless $% s_{x^2 y^2} $-wave through
the intermediate $s+id\times s$ mixed pairing phase, which is consistent with
the observation of pressure experiments. The emergent $d$-wave form factor
inevitably arises from the projection of inter-orbital Cooper pairing onto the
Fermi surface and is inherited from the electronic structure in the
representation of $C_{4}^{p}$ symmetry. Moreover, the $S_{4}$ symmetry dictates
2 copies of $d$-wave pairing condensates, counting 8 nodes in total. We further
show that weakly breaking $C_{4}^{p}$ naturally leads to the octet nodal gap as
precisely observed in laser angle resolved photoemission spectroscopy. The
octet nodes reflect the collaboration of the $C_{4}^{p}$ and $S_{4}$
symmetries, which sheds new light on the enigma of the pairing symmetry in
$\text{KFe}_{2}\text{As}_{2}$.
