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