In his 1984 AMS Memoir, Andrews introduced the notion of \(c\phi _k(n)\) , which is the number of k-colored generalized Frobenius partitions of n. In 2019, by using the theory of modular forms, Chan, Wang and Yang considered the arithmetic properties of \(\text {C}\Phi _k(q)\) for \(2\le k\le 17\) , where \(\text {C}\Phi _k(q)\) denotes the generating function of \(c\phi _k(n)\) . More recently, the first author, Gu and Tang obtained the generating functions for \(\text {C}\Phi _{12}(q)\) and derived some congruence properties for \(c\phi _k(n)\) modulo \(3^3\) and \(3^4\) . In this paper, we obtain a new expression of \(\text {C}\Phi _{12}(q)\) . Meanwhile, some new congruences for \(c\phi _{12}(n)\) modulo powers of 3 are established.
