GAP (ClassicalMaximals) - Chapter 4: Examples

The following examples illustrate the use of ClassicalMaximalsGeneric (3.1-1).

Consider the maximal subgroups of \({\rm SU}_5(7)\). These can be obtained directly as follows:

gap> L := ClassicalMaximalsGeneric("U", 5, 7);
[ <matrix group of size 223883102951424 with 6 generators>,
  <matrix group of size 32541148684800 with 6 generators>,
  <matrix group of size 37298309529600 with 4 generators>,
  <matrix group of size 15223799808 with 5 generators>,
  <matrix group of size 491520 with 4 generators>,
  <matrix group of size 10505 with 2 generators>,
  <matrix group of size 276595200 with 4 generators>,
  <matrix group of size 660 with 2 generators> ]
gap> DefaultFieldOfMatrixGroup(L[1]);
GF(7^2)

Note that unitary groups are defined over \(\mathbb{F}_{q^2}\), even though they are parametrised by \(q\) (see 2.2-3).

It is often useful to restrict attention to certain Aschbacher classes. For example, the reducible and imprimitive maximal subgroups of \({\rm Sp}_6(9)\) (that is, classes \({\cal C}_1\) and \({\cal C}_2\)) can be obtained by specifying the optional argument classes:

gap> ClassicalMaximalsGeneric("S", 6, 9, [1, 2]);
[ <matrix group of size 1626546181017600 with 6 generators>,
  <matrix group of size 19835929036800 with 6 generators>,
  <matrix group of size 180506954234880 with 3 generators>,
  <matrix group of size 2479113216000 with 4 generators>,
  <matrix group of size 2239488000 with 4 generators>,
  <matrix group of size 679311360 with 3 generators> ]

The groups returned by ClassicalMaximalsGeneric (3.1-1) are realised as subgroups of the corresponding standard classical group and therefore preserve the standard form. This can be verified using the stored invariant forms. For example, consider a maximal subgroup of \({\rm \Omega}^-_8(5)\) in class \({\cal C}_9\):

gap> G := ClassicalMaximalsGeneric("O-", 8, 5, [9])[1];
<matrix group of size 372000 with 2 generators>
gap> Display(InvariantBilinearForm(G).matrix);
 . . . . . . . 1
 . . . . . . 1 .
 . . . . . 1 . .
 . . . 3 4 . . .
 . . . 4 1 . . .
 . . 1 . . . . .
 . 1 . . . . . .
 1 . . . . . . .
gap> Display(InvariantBilinearForm(Omega(-1, 8, 5)).matrix);
 . . . . . . . 1
 . . . . . . 1 .
 . . . . . 1 . .
 . . . 3 4 . . .
 . . . 4 1 . . .
 . . 1 . . . . .
 . 1 . . . . . .
 1 . . . . . . .

The two matrices coincide, confirming that the subgroup preserves our standard orthogonal form.

4 Examples