groups – The symmetric group Sn contains, for any group of order n, a subgroup isomorphic to that group. Thus the cycle graph of every group of order n will be found in the cycle graph of Sn. See example: Subgroups of S4 Example: Subgroups of the full octahedral group The full octahedral group is the direct product of the symmetric group S4 and the cyclic group Z2. Its order is 48, and it has subgroups of every order that divides 48. In the examples below nodes that are related to each other are placed next to each other, so these are not the simplest possible cycle graphs for these groups (like those on the right). Like all graphs a cycle graph can be represented in different ways to emphasize different properties. The two representations of the cycle graph of S4 are an example of that. See also List of small groups Cayley graph External links References Skiena, S. (1990). Cycles, Stars, and Wheels. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (pp. 144-147). Pemmaraju, S., & Skiena, S. (2003). Cycles, Stars, and Wheels. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (pp. 248-249). Cambridge University Press. Abstract algebra Group theory Application-specific graphs