The symplectic group $Sp\left(n\,q\right)$ is the group of all $n\times n$ matrices over the field with $q$ elements that respect a fixed nondegenerate symplectic form. The integer $n$ must be even.

The SymplecticGroup( n, q ) command returns a permutation group isomorphic to the symplectic group $Sp\left(n\,q\right)$ .

Note that for $n=2$ the groups $Sp\left(n\,q\right)$ and $SL\left(n\,q\right)$ are isomorphic, so that a special linear group is returned in this case.

The implemented ranges for $n$ and $q$ are as follows:

If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SymplecticGroup}\left(4\,5\right)$

$G≔\mathbf{Sp}\left(4\,5\right)$

$\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$

${\left(2\right)}^7{\left(3\right)}^2{\left(5\right)}^4\left(13\right)$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(2\,G\right)\right)$

$128$

$\mathrm{S3}≔\mathrm{SylowSubgroup}\left(3\,G\right)$

$\mathrm{S3}≔\mathrm{<a\; permutation\; group\; on\; 624\; letters>}$

$\mathrm{GroupOrder}\left(\mathrm{S3}\right)$

$9$

$\mathrm{IsCyclic}\left(\mathrm{S3}\right)$

$\mathrm{false}$

$\mathrm{IdentifySmallGroup}\left(\mathrm{S3}\right)$

$9,2$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(5\&comma;G\right)\right)$

$625$

$\mathrm{IsTrivial}\left(\mathrm{PCore}\left(5\&comma;G\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(13\&comma;G\right)\right)$

$13$

$G≔\mathrm{SymplecticGroup}\left(4\&comma;3\right)$

$G≔\mathbf{Sp}\left(4\&comma;3\right)$

$\mathrm{Degree}\left(G\right)$

$80$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(\mathrm{Centre}\left(G\right)\right)$

$2$

$\mathrm{SymplecticGroup}\left(2\&comma;5\right)$

$\mathbf{SL}\left(2\&comma;5\right)$

$\mathrm{AreIsomorphic}\left(\mathrm{SymplecticGroup}\left(4\&comma;2\right)\&comma;\mathrm{Symm}\left(6\right)\right)$

$\mathrm{true}$

$G≔\mathrm{SymplecticGroup}\left(6\&comma;q\right)$

$G≔\mathbf{Sp}\left(6\&comma;q\right)$

$\mathrm{GroupOrder}\left(G\right)$

$q^9\left(q^2-1\right)\left(q^4-1\right)\left(q^6-1\right)$

$\mathrm{ClassNumber}\left(\mathrm{SymplecticGroup}\left(8\&comma;q\right)\right)$

$\left\{\begin{array}{cc}5q+\left(q+1\right)q+4q^2+\left(q^2+q+3\right)q+q^4+q^3+7& q::\mathrm{even}\\ 25q+51+\left(q+4\right)q+11q^2+\left(q^2+4q+10\right)q+q^4+4q^3& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ClassNumber}\left(\mathrm{SymplecticGroup}\left(4\&comma;{11}^k\right)\right)\mathbf{assuming}k::\&apos;\mathrm{posint}\&apos;$

$5{11}^k+10+{\left({11}^k\right)}^2$

The GroupTheory[SymplecticGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[SymplecticGroup] command was updated in Maple 2020.