Database of Finite Quaternionic Matrix Groups

Magma includes a database of the finite absolutely irreducible subgroups of GLn((D)) where (D) is a definite quaternion algebra whose centre has degree d over Q and nd leq10. This collection is due to Gabriele Nebe [Neb98]. This section defines the interface to that database.

A particular entry of the database can be specified in one of two ways. Firstly, a number in the range 1 to the size of the database can be given. Alternatively, the desired dimension can be provided, together with a number in the range 1 to the number of entries of that dimension.

Each entry can be accessed either as a matrix group or as a lattice. If accessed as a matrix group, the order and base are set on return.

QuaternionicMatrixGroupDatabase() : -> DB
This function returns a database object which contains information about the database.
LargestDimension(D) : DB -> RngIntElt
Returns the largest dimension of any entry stored in the database. It is an error to refer to larger dimensions in the database.
# D : DB -> RngIntElt
NumberOfGroups(D) : DB -> RngIntElt
NumberOfLattices(D) : DB -> RngIntElt
Returns the number of entries stored in the database.
NumberOfGroups(D, d) : DB, RngIntElt -> RngIntElt
NumberOfLattices(D, d) : DB, RngIntElt -> RngIntElt
Returns the number of entries stored in the database of dimension d.
Group(D, i): DB, RngIntElt -> GrpMat
Returns the i-th entry from the database D as a matrix group.
Lattice(D, i): DB, RngIntElt -> Lat, SeqEnum
Returns a lattice L and sequence of forms F corresponding to the i-th entry of the database D.
Construction(D, i): DB, RngIntElt -> MonStgElt, RngIntElt
Returns a string and integer which describe the construction of the i-th entry of the database D.
Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat
Returns the i-th entry of dimension d in the database D as a matrix group.
Lattice(D, d, i): DB, RngIntElt, RngIntElt -> Lat, SeqEnum
Returns a lattice L and sequence of forms F corresponding to the i-th entry of dimension d in the database D.
Construction(D, d, i): DB, RngIntElt, RngIntElt -> MonStgElt, RngIntElt
Returns a string and integer which describe the construction of the i-th entry of dimension d in the database D.

Example GrpData_Quaternionic (H72E18)

We illustrate accessing the quaternionic matrix groups database with a group and lattice of dimension 36. 
> DB := QuaternionicMatrixGroupDatabase();
> LargestDimension(DB);
40
> NumberOfGroups(DB, 36);
10
> G := Group(DB, 36, 8);
> G : Minimal;
MatrixGroup(36, Integer Ring) of order 43545600 = 2^10 * 3^5
* 5^2 * 7
> #pCore(G, 2);
2
> L, forms := Lattice(DB, 36, 8);
> Determinant(L);
3874204890000
> IsSquare($1);
true 1968300
V2.28, 13 July 2023