  Single crystals of Frank-Kasper compounds RTM$_2$Al$_{20}$ (R = Eu, Gd and
La; TM = V and Ti) were grown by self-flux method and their physical properties
were investigated through magnetization ($M$), magnetic susceptibility
($\chi$), specific heat ($C_P$) and electrical resistivity ($\rho$)
measurements. Powder x-ray diffraction studies and structural analysis showed
that these compounds crystallize in the cubic crystal structure with the space
group $Fd\overline{3}m$. The magnetic susceptibility for the compounds
EuTi$_2$Al$_{20}$ and GdTi$_2$Al$_{20}$ showed a sudden jump below the N{\'e}el
temperature $T_N$ indicative of plausible double magnetic transition. Specific
heat ($C_P$) and electrical resistivity ($\rho$) measurements also confirm the
first-order magnetic transition (FOMT) and possible double magnetic
transitions. Temperature variation of heat capacity showed a sharp phase
transition and huge $C_P$ value for the (Eu/Gd)Ti$_2$Al$_{20}$ compounds Full
width at half-maximum (FWHM) $<$ 0.2 K) which is reminiscent of a first-order
phase transition and a unique attribute among RTM$_2$Al$_{20}$ compounds. We
observed clear anomaly between heating and cooling cycle in temperature-time
relaxation curve for the compounds GdTi$_2$Al$_{20}$ (2.38 $K$) and
EuTi$_2$Al$_{20}$ (3.2 $K$) which is indicating a thermal arrest due to the
latent heat. The temperature variation of $S_{mag}$ for GdTi$_2$Al$_{20}$
saturates to a value $0.95R\ln8$ while the other magnetic systems exhibited
still lower entropy saturation values in the high temperature limit.
Resistivity measurements showed that all the samples behave as normal Fermi
liquid type compounds and $\rho(T)$ due to electron-phonon scattering follows
Bloch-Gr$\ddot{\textrm u}$neisen-Mott relation in the paramagnetic region.
