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In this paper we extend the study of three state lively quantum walks oncycles by considering the coin operator as a linear sum of permutationmatrices, which is a generalization of the Grover matrix. First we provide acomplete characterization of orthogonal matrices of order $3\times 3$ which arelinear sum of permutation matrices. Consequently, we determine several groupsof complex, real and rational orthogonal matrices. We establish that anorthogonal matrix of order $3\times 3$ is a linear sum of permutation matricesif and only if it is permutative. Finally we determine period of lively quantumwalk on cycles when the coin operator belongs to the group of orthogonal (real)linear sum of permutation matrices.

Periodicity of lively quantum walks on cycles with generalized Grover coin