The symplectic semi-linear group $\Sigma p\left(n\,q\right)$ is the set of all semi-linear transformations of an $n$-dimensional vector space $V$ over the field with $q$ elements whose linear part preserves a non-degenerate symplectic form. The dimension $n$ must be an even positive integer. The group $\Sigma p\left(n\,q\right)$ is a semi-direct product of the symplectic group $Sp\left(n\,q\right)$ with the Galois group of the field GF(q). Therefore, if $q$ is prime, $\Sigma p\left(n\,q\right)$ is isomorphic to $Sp\left(n\,q\right)$ . Furthermore, if $n=2$, then $Sp\left(n\,q\right)$ and $SL\left(n\,q\right)$ coincide, so $\Sigma L\left(2\,q\right)$ is returned in this case.

If n and q are positive integers, then the SymplecticSemilinearGroup( n, q ) command returns a permutation group isomorphic to the symplectic semi-linear group  $\Sigma p\left(n\,q\right)$ . Otherwise, a symbolic group is returned, with which Maple can do some limited computations.

The abbreviation Sigmap( n, q ) is available as a synonym for SymplecticSemilinearGroup( n, q ).

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

$G≔\mathrm{SymplecticSemilinearGroup}\left(2\,8\right)$

$G≔\mathbf{\Sigma L}\left(2\,8\right)$

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

$1512$

$\mathrm{GroupOrder}\left(\mathrm{SigmaL}\left(2\,8\right)\right)$

$1512$

$G≔\mathrm{Sigmap}\left(4\,4\right)$

$G≔\mathbf{\Sigma p}\left(4\,4\right)$

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

$1958400$

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

$\mathrm{false}$

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

$\mathrm{true}$

$G≔\mathrm{Sigmap}\left(8\,q\right)$

$G≔\mathbf{\Sigma p}\left(8\,q\right)$

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

$\mathrm{logp}\left(q\right)q^{16}\left(q^2-1\right)\left(q^4-1\right)\left(q^6-1\right)\left(q^8-1\right)$