Database of Finite Quaternionic Matrix Groups

Magma includes 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 \(nd\leq10\). This collection is due to Gabriele Nebe [Nebe, 1998]. 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: Quaternionic (ex-00c791)

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

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