A cycle graph of a group is a graph which shows cycles of a group as well as the connectivity
between the cycles. Such graphs are constructed by drawing labeled nodes, one for
each element 
of the group, and connecting cycles obtained by iterating 
. Each edge of such a graph is bidirected, but they are commonly
drawn using undirected edges with double edges used to indicate cycles of length
two (Shanks 1993, pp. 85 and 87-92). Cycle graphs are generally drawn without
a self-loop from the identity element to itself, but also without any implicit subcycles.
For instance, the cycle graph of the cyclic group 
is drawn as a single closed loop of
length 8 produced by the generator 
, omitting the period-2 and 4 subcycles and the cycles generated
by 
and 
.
Several examples are shown above.
The cycle graph of the cyclic group 
consists of a loop connecting the 
group elements. The cycle graph of the dihedral
group 
consists of a loop connecting 
elements together with 
lobes (2-cycles) sticking out from the node representing the
identity element.
Precomputed cycle graphs for a number of finite groups are available in the Wolfram Language using FiniteGroupData[gr, "CycleGraph"].
