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.