The projective general orthogonal group $PGO\left(d\,n\,q\right)$ is the quotient of the general orthogonal group $GO\left(d\,n\,q\right)$ by its center. The value of $d$ must be $0$ for odd $n$, or $1$ or $\mathrm{-1}$ for even $n$.

The ProjectiveGeneralOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the general orthogonal group $GO\left(d\,n\,q\right)$ .

The PGO( d, n, q ) command is provided as an alias.

If the argument q is not a prime power (and is non-numeric), then a symbolic group representing $PGO\left(d\,n\,q\right)$ is returned.

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

$G≔\mathrm{ProjectiveGeneralOrthogonalGroup}\left(0\,3\,3\right)$

$G≔\mathbf{PGO}\left(0\,3\,3\right)$

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

$24$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsDihedral}\left(\mathrm{PGO}\left(-1\,2\,8\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{PGO}\left(0\,5\,q\right)\right)$

$\left\{\begin{array}{cc}\frac{\mathrm{igcd}\left(2\,q-1\right)q^4\left(q^2-1\right)\left(q^4-1\right)}{2}& q::\mathrm{odd}\\ \mathrm{igcd}\left(2\,q-1\right)q^4\left(q^2-1\right)\left(q^4-1\right)& \mathrm{otherwise}\end{array}\right.$

$\mathrm{IsTrivial}\left(\mathrm{PGO}\left(0\,1\,{31}^{10}\right)\right)$

$\mathrm{true}$