  Members of the CaK(Fe$_{1-x}$Cr$_{x}$)$_{4}$As$_{4}$ series have been
synthesized by high-temperature solution growth in single crystalline form and
characterized by X-ray diffraction, elemental analysis, magnetic and transport
measurements. The effects of Cr substitution on the superconducting and
magnetic ground states of CaKFe$_4$As$_4$ ($T_c$ = 35 K) have been studied.
These measurements show that the superconducting transition temperature
decreases monotonically and is finally suppressed below 1.8 K as $x$ is
increased from 0 to 0.038. The magnetic transition temperature increases in a
roughly linear manner as Cr substitution increases. A temperature-composition
(\textit{T}-\textit{x}) phase diagram is constructed, revealing a half-dome of
superconductivity with the magnetic transition temperature, $T^*$, appearing
near 22~K for $x$ $\sim$ 0.017 and rising slowly up to 60~K for $x$ $\sim$
0.077. The $T$-$x$ phase diagrams for CaK(Fe$_{1-x}$$T$$_{x}$)$_4$As$_4$ for
$T$ = Cr and Mn are essentially the same despite the nominally different band
filling; this is in marked contrast to $T$ = Co and Ni series for which the
$T$-$x$ diagrams scale by a factor of two, consistent with the different
changes in band filling Co and Ni would produce when replacing Fe.
Superconductivity of CaK(Fe$_{1-x}$Cr$_{x}$)$_{4}$As$_{4}$ is also studied as a
function of magnetic field. A clear change in $H^\prime_{c2}$($T$)/$T_c$, where
$H^\prime_{c2}$($T$) is d$H_{c2}$($T$)/d$T$, at $x$ $\sim$ 0.012 is observed
and probably is related to change of the Fermi surface due to magnetic order.
Coherence length and the London penetration depths are also calculated based on
$H_{c1}$ and $H_{c2}$ data. Coherence lengths as the function of $x$ also shows
changes near $x$ = 0.012, again consistent with Fermi surfaces changes
associated with the magnetic ordering seen for higher $x$-values.
