We report a study of the atomic and electronic structures of the ordered ${\mathrm{Ag}}_{2}\mathrm{Ge}$ surface alloy containing ⅓ monolayer of Ge. Low-energy electron diffraction (LEED), scanning tunneling microscopy (STM), and angle-resolved photoelectron spectroscopy (ARPES) data reveal a symmetry breaking of the expected \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 periodicity, which is established for other ${\mathrm{Ag}}_{2}$$M$ alloys ($M$ = Bi, Sb, Pb, and Sn). The deviation from a simple \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 structure manifests itself as a splitting of diffraction spots in LEED, as a striped structure with a 6\ifmmode\times\else\texttimes\fi{} periodicity including a distortion of the local hexagonal structure in STM, and as a complex surface band structure in ARPES that is quite different from those of the other ${\mathrm{Ag}}_{2}$$M$ alloys. These results are interesting in view of the differences in the atomic and electronic structures exhibited by different group IV elements interacting with Ag(111). Pb and Sn form \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 surface alloys on Ag(111), of which ${\mathrm{Ag}}_{2}\mathrm{Pb}$ shows a surface band structure with a clear spin-orbit split. Si and C form silicene and graphene structures, respectively, with linear band dispersions and the formation of Dirac cones as reported for graphene. The finding that ${\mathrm{Ag}}_{2}\mathrm{Ge}$ deviates from the ideal (\ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3) ${\mathrm{Ag}}_{2}\mathrm{Sn}$ and ${\mathrm{Ag}}_{2}\mathrm{Pb}$ surface alloys makes Ge an interesting ``link'' between the heavy group IV elements (Sn, Pb) and the light group IV elements (Si, C).