Reproducing quadruped locomotion from an engineering viewpoint is important not only to control robot locomotion but also to clarify the nonlinear mechanism for switching between locomotion patterns. In this paper, we reproduced a quadruped locomotion pattern, gallop, using a central pattern generator (CPG) hardware network based on the abelian group Z₄×Z₂, originally proposed by Golubitsky et al. We have already used the network to generate three locomotion patterns, walk, trot, and bound, by controlling the voltage, E_MLR, inputted to all CPGs which acts as a signal from the midbrain locomotor region (MLR). In order to generate the gallop and canter patterns, we first analyzed the network symmetry using group theory. Based on the results of the group theory analysis, we desymmetrized the contralateral couplings of the CPG network using a new parameter in addition to E_MLR, because, whereas the walk, trot, and bound patterns were able to be generated from the spatio-temporal symmetry of the product group Z₄×Z₂, the gallop and canter patterns were not. As a result, using a constant element $\hat{\kappa}$ on Z₂, the gallop and canter locomotion patterns were generated by the network on ${\bf Z}_4+\hat{\kappa}{\bf Z}_4$, and actually in this paper, the gallop locomotion pattern was generated on the actual circuit.