In this paper, we present a new algebraic framework that provides an effective approach to investigate controllability of systems defined on the special orthogonal group. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications in a symmetric group, so that controllability and controllable submanifolds can be characterized by permutation cycles. This new notion enables a visualization of controllability analysis over an undirected graph and facilitates the design of efficient computational algorithms to examine controllability and identify the controllable submanifold by computing permutation cycles. Furthermore, the developed methodology reveals the relationship between controllability of a system and connectivity of the associated graph, which renders a transparent way to understand controllability over graphs. The method is directly applicable to characterize the degree of controllability and reachability of systems defined on compact Lie groups and on graphs, such as quantum networks, multi-agent systems, and Markov chains.