The symmetry requirement and the origin of magnetic orders coexisting with superconductivity have been strongly debated issues of iron-based superconductors (FeSCs). Observation of C$_4$-symmetric antiferromagnetism in violation of the inter-band nesting condition of spin-density waves in superconducting ground state will require significant change in our understanding of the mechanism of FeSC. The superconducting material Sr$_2$VO$_3$FeAs, a bulk version of monolayer FeSC in contact with a perovskite layer with its magnetism (T$_N$ ~ 50 K) and superconductivity (T$_c$ ~ 37 K) coexisting at parent state, has no reported structural orthorhombic distortion and thus makes a perfect system to look for theoretically expected C$_4$ magnetisms. Based on variable temperature spin-polarized scanning tunneling microscopy (SPSTM) with newly discovered imaging mechanism that removes the static surface reconstruction (SR) pattern by fluctuating it rapidly with spin-polarized tunneling current, we could visualize underlying C$_4$ symmetric (2$times$2) magnetic domains and its phase domain walls. We find that this magnetic order is perfectly consistent with the plaquette antiferromagnetic order in tetragonal Fe spin lattice expected from theories based on the Heisenberg exchange interaction of local Fe moments and the quantum order by disorder. The inconsistency of its modulation Q vectors from the nesting condition also implies that the nesting-based C$_2$ symmetric magnetism is not a unique prerequisite of high-T$_c$ FeSC. Furthermore, the plaquette antiferromagnetic domain wall dynamics under the influence of small spin torque effect of spin-polarized tunneling current are shown to be consistent with theoretical simulation based on the extended Landau-Lifshitz-Gilbert equation.
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