There are a number of ways that we can specify and work with groups.

- Cayley Table - This is the multiplication table for the group. This completely defines the group although many of the table entries can derived from the others so it contains quite a lot of redundancy.
- Cayley Graph - This shows the group in terms of the group generator(s). There may be some freedom of choice in selecting the generators for a given group.
- Cyclic Notation (or cycle graph) - This lists the group elements with the cycles, for a given permutation, grouped together using brackets.
- Group Presentation - Like the Cayley Graph this defines the group in terms of a set of generators but then defines constraints on these generators using equations.
- Group Representation- Representation theory represents the group as a morphism of the group onto linear transformations (matrices or tensors)

There are also other diagrams which don't necessarily define the group but show the relationship between groups.

- Hasse Diagrams - This shows all the subgroups of a given group and their relationship.