The $\ensuremath{\Delta}I=1$ band built on the $5645\mathrm{keV}$ ${(14}^{\ensuremath{-}})$ level in ${}^{136}\mathrm{Ce}$ has been extended to the $9228\mathrm{keV}$ ${(23}^{\ensuremath{-}})$ level using an eight CS-Clover array with $14$ element NaI(Tl) multiplicity filter. The crossover $E2$ transitions have been identified for the first time. The lifetimes of the ${18}^{\ensuremath{-}}$ to ${22}^{\ensuremath{-}}$ levels have also been measured by the Doppler shift attenuation method. The observed band is reproduced well by the ``tilted axis cranking''(TAC) calculations for the four quasiparticle $\ensuremath{\nu}{h}_{11/2}^{\ensuremath{-}2}\ensuremath{\bigotimes}\ensuremath{\pi}{g}_{7/2}{h}_{11/2}$ configuration having a small oblate deformation $({\ensuremath{\epsilon}}_{2}\ensuremath{\sim}0.11)$ and a tilt angle of $32\ifmmode^\circ\else\textdegree\fi{}.$ The measured $B(M1,\stackrel{\ensuremath{\rightarrow}}{I}I\ensuremath{-}1)/B(E2,\stackrel{\ensuremath{\rightarrow}}{I}I\ensuremath{-}2)$ ratios as well as the $B(M1,\stackrel{\ensuremath{\rightarrow}}{I}I\ensuremath{-}1)$ values for ${18}^{\ensuremath{-}}$ to ${23}^{\ensuremath{-}}$ levels show a decrease with increasing rotational frequency in agreement with these calculations, thus confirming the ``magnetic rotation'' (MR) nature of the band. The calculations also reveal the presence of a second minimum in energy for the four quasiparticle $\ensuremath{\nu}{h}_{11/2}^{\ensuremath{-}2}\ensuremath{\bigotimes}\ensuremath{\pi}{g}_{7/2}{h}_{11/2}$ configuration having triaxial shape with $\ensuremath{\gamma}=21\ifmmode^\circ\else\textdegree\fi{}.$ This minimum also supports a closely lying band with a tilt angle of $82\ifmmode^\circ\else\textdegree\fi{},$ implying principal axis cranking (PAC). The measured values of $B(M1,\stackrel{\ensuremath{\rightarrow}}{I}I\ensuremath{-}1)/B(E2,\stackrel{\ensuremath{\rightarrow}}{I}I\ensuremath{-}2)$ ratios, which are quite small at the lower spins, are also in agreement with the calculated values for the triaxial shape and a mixing with the near oblate shape. The sudden rise in the $B(M1)/B(E2)$ values at $I=18$ clearly indicates a transition from the PAC regime to magnetic rotation, which is induced by the shape change.