  The magnetic order for several compositions of
CaK(Fe$_{1-x}$Mn$_x$)$_4$As$_4$ has been studied by nuclear magnetic resonance
(NMR), M\"ossbauer spectroscopy, and neutron diffraction. Our observations for
the Mn-doped 1144 compound are consistent with the hedgehog spin vortex crystal
(hSVC) order which has previously been found for Ni-doped
$\text{Ca}\text{K}\text{Fe}_4\text{As}_4$. The hSVC state is characterized by
the stripe-type propagation vectors $(\pi\,0)$ and $(0\,\pi)$ just as in the
doped 122 compounds. The hSVC state preserves tetragonal symmetry at the Fe
site, and only this SVC motif with simple AFM stacking along $\textbf{c}$ is
consistent with all our observations using NMR, M\"ossbauer spectroscopy, and
neutron diffraction. We find that the hSVC state in the Mn-doped 1144 compound
coexists with superconductivity (SC), and by combining the neutron scattering
and M\"ossbauer spectroscopy data we can infer a quantum phase transition,
hidden under the superconducting dome, associated with the suppression of the
AFM transition temperature ($T_\text{N}$) to zero for $x\approx0.01$. In
addition, unlike several 122 compounds and Ni-doped 1144, the ordered magnetic
moment is not observed to decrease at temperatures below the superconducting
transition temperature ($T_\text{c}$).
