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Molecular mobility and conductivity of four bent shaped tetramethylated guanidinium based ionic liquid crystals (ILCs) with varying head group configuration (cyclic or acyclic) and alkyl chain length is investigated by a combination of broadband dielectric spectroscopy (BDS) and specific heat spectroscopy (SHS). Two dielectrically active processes observed in the plastic crystalline phase at low and high temperatures are denoted as γ and α1 relaxation. The former is assigned to localized fluctuations of methyl groups including nitrogen atoms in the guanidinium head groups. SHS investigations reveal one calorimetrically active process termed as α2 relaxation process. The temperature dependencies of the relaxation rates of α1 and α2 are similar for the cyclic ILC while for the acyclic counterpart they are different. Possible molecular assignments for the α1 and α2 relaxation are discussed in detail. Alongside relaxation processes, a significant conductivity contribution was observed for all ILCs, where the absolute value of DC conductivity increases by 4 orders of magnitude at the transition from the crystalline to the hexagonal columnar phase. The increase is traced to the change in the underlying conduction mechanism from the delocalized electrical conduction in the Cry phase to ionic conduction in the quasi 1D ion columns formed in the hexagonal columnar mesophase.