  Cobalt orthogermanate (GeCo$_2$O$_4$) is a unique system in the family of
cobalt spinels ACo$_2$O$_4$ (A= Sn, Ti, Ru, Mn, Al, Zn, Fe, etc.) in which
magnetic Co ions stabilize on the pyrochlore lattice exhibiting a large degree
of orbital frustration. Due to the complexity of the low-temperature
antiferromagnetic (AFM) ordering and long-range magnetic exchange interactions,
the lattice dynamics and magnetic structure of GeCo$_2$O$_4$ spinel has
remained puzzling. To address this issue, here we present theoretical and
experimental investigations of the highly frustrated magnetic structure, and
the infrared (IR) and Raman-active phonon modes in the spinel GeCo$_2$O$_4$,
which exhibits an AFM ordering below the N\'eel temperature $T_N$ ~21 K,
followed by a cubic ($Fd{\bar 3}m$) to tetragonal ($I4_{1}/amd$) structural
phase transition at $T_S$ ~16 K. Our density-functional theory (DFT+U)
calculations reveal that one needs to consider magnetic-exchange interactions
up to the third nearest neighbors to get an accurate description of the
low-temperature AFM order in GeCo$_2$O$_4$. At room temperature three distinct
IR-active modes ($T_{1u}$) are observed at frequencies 680, 413, and 325
cm$^{-1}$ along with four Raman-active modes $A_{1g}$, $T_{2g}(1)$,
$T_{2g}(2)$, and $E_{g}$ at frequencies 760, 647, 550, and 308 cm$^{-1}$,
respectively, which match reasonably well with our DFT+U calculated values. All
the IR-active and Raman-active phonon modes exhibit signatures of moderate
spin-phonon coupling. The temperature dependence of various parameters, such as
the shift, width, and intensity, of the Raman-active modes, is also discussed.
Noticeable changes around $T_N$ and $T_S$ are observed in the Raman line
parameters of the $E_{g}$ and $T_{2g}$ modes, which are associated with the
modulation of the Co-O bonds in CoO$_6$ octahedra during the excitations of
these modes.
