# Databases of Groups

Magma contains the following databases of groups:

*Small Groups*: Contains all groups of order up to 1000, excluding orders 512 and 768.

*Perfect Groups*: This database contains all perfect groups up to order 50000, and many classes of perfect groups up to order one million. Each group is defined by means of a finite presentation. Further information is also provided which allows the construction of permutation representations.

*Rational Maximal Matrix Groups*: Contains rational maximal finite matrix groups and their invariant forms, for small dimensions (up to 31 at V2.9 and above). Each entry can be accessed either as a matrix group or as a lattice.

*Quaternionic Matrix Groups*: A database of the finite absolutely irreducible subgroups of ${\operatorname{GL}}_n({\cal D})$ where ${\cal D}$ is a definite quaternion algebra whose centre has degree $d$ over ${\mathbb{Q}}$ and $nd\leq10$. Each entry can be accessed either as a matrix group or as a lattice.

*Transitive Permutation Groups*: Magma has a database containing all transitive permutation groups having degree up to 47.

*Primitive Permutation Groups*: Magma has a database containing all primitive permutation groups having degree up to 4095.

For a description of these databases, we refer to Chapter [Databases of Groups](../DatabasesOfGroups/index-databases-of-groups.md#chapgrpdata).