The general orthogonal group $GO\left(d\,n\,q\right)$ is the set of all $n\times n$ matrices over the field with $q$ elements that respect a non-singular quadratic form. The value of $d$ must be $0$ for odd $n$, or $1$ or $\mathrm{-1}$ for even $n$.

The GeneralOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the general orthogonal group $GO\left(d\,n\,q\right)$ for the implemented values of $d$, $n$ and $q$.

The implemented ranges for n and q are as follows:

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

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

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

$G≔\mathrm{GeneralOrthogonalGroup}\left(0\,7\,2\right)$

$G≔\mathbf{GO}\left(7\,2\right)$

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

$\left[\left(1\,2\,3\,4\,5\,7\right)\left(6\,8\,11\,15\,21\,27\right)\left(9\,12\,16\,22\,29\,38\right)\left(10\,13\,18\,23\,31\,41\right)\left(14\,19\,24\,33\,44\,34\right)\left(17\,20\,25\right)\left(26\,35\,45\,53\,32\,42\right)\left(28\,36\,47\right)\left(30\,39\,50\,56\,61\,58\right)\left(37\,48\right)\left(40\,51\,46\,54\,59\,62\right)\left(49\,55\,60\,63\,52\,57\right)\,\left(5\,6\right)\left(7\,9\right)\left(8\,10\right)\left(11\,14\right)\left(13\,17\right)\left(15\,20\right)\left(18\,19\right)\left(21\,26\right)\left(22\,28\right)\left(23\,30\right)\left(24\,32\right)\left(25\,34\right)\left(29\,37\right)\left(31\,40\right)\left(33\,43\right)\left(36\,46\right)\left(38\,41\right)\left(39\,49\right)\left(42\,44\right)\left(45\,52\right)\left(48\,51\right)\left(53\,58\right)\left(57\,62\right)\left(59\,61\right)\right]$

$G≔\mathrm{GeneralOrthogonalGroup}\left(1\,4\,5\right)$

$G≔\mathbf{GO}\left(4\,5\right)$

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

$28800$

$G≔\mathrm{GeneralOrthogonalGroup}\left(-1\,4\,5\right)$

$G≔\mathbf{GO}\left(4\,5\right)$

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

$104$

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

$31200$

$G≔\mathrm{GeneralOrthogonalGroup}\left(0\,3\,5\right)$

$G≔\mathbf{GO}\left(3\,5\right)$

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

$\begin{array}{c}\left[\begin{array}{ccccccccccc}C& \mathrm{1a}& \mathrm{2a}& \mathrm{2b}& \mathrm{2c}& \mathrm{2d}& \mathrm{2e}& \mathrm{3a}& \mathrm{4a}& \mathrm{4b}& \dots \\ \mathrm{|C|}& 1& 1& 10& 10& 15& 15& 20& 30& 30& \dots \\ \mathrm{X1}& 1& 1& 1& 1& 1& 1& 1& 1& 1& \dots \\ \mathrm{X2}& 1& \mathrm{-1}& \mathrm{-1}& 1& 1& \mathrm{-1}& 1& \mathrm{-1}& 1& \dots \\ \mathrm{X3}& 1& \mathrm{-1}& 1& \mathrm{-1}& 1& \mathrm{-1}& 1& 1& \mathrm{-1}& \dots \\ \mathrm{X4}& 1& 1& \mathrm{-1}& \mathrm{-1}& 1& 1& 1& \mathrm{-1}& \mathrm{-1}& \dots \\ \mathrm{X5}& 4& \mathrm{-4}& \mathrm{-2}& 2& 0& 0& 1& 0& 0& \dots \\ \mathrm{X6}& 4& \mathrm{-4}& 2& \mathrm{-2}& 0& 0& 1& 0& 0& \dots \\ \mathrm{X7}& 4& 4& \mathrm{-2}& \mathrm{-2}& 0& 0& 1& 0& 0& \dots \\ \mathrm{X8}& 4& 4& 2& 2& 0& 0& 1& 0& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{16 × 15 Matrix}\end{array}$

$\mathrm{OrderClassPolynomial}\left(G\,x\right)$

$24x^{10}+60x^6+24x^5+60x^4+20x^3+51x^2+x$

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

$\mathbf{GO}\left(3\,5\right)▹⟨\left(1\,4\right)\left(2\,8\right)\left(3\,9\right)\left(5\,15\right)\left(6\,23\right)\left(7\,20\right)\left(10\,12\right)\left(11\,19\right)\left(13\,16\right)\left(14\,24\right)\left(17\,21\right)\left(18\,22\right)\,\left(1\,4\right)\left(2\,8\right)\left(3\,22\right)\left(5\,6\right)\left(7\,18\right)\left(9\,10\right)\left(11\,23\right)\left(12\,17\right)\left(13\,14\right)\left(15\,16\right)\left(19\,24\right)\left(20\,21\right)\,\left(1\,20\right)\left(2\,19\right)\left(3\,9\right)\left(4\,21\right)\left(5\,15\right)\left(6\,16\right)\left(7\,12\right)\left(8\,24\right)\left(10\,22\right)\left(11\,13\right)\left(14\,23\right)\left(17\,18\right)⟩$

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

$⟨\left(1\,4\right)\left(2\,8\right)\left(3\,9\right)\left(5\,15\right)\left(6\,16\right)\left(7\,17\right)\left(10\,22\right)\left(11\,14\right)\left(12\,18\right)\left(13\,23\right)\left(19\,24\right)\left(20\,21\right)⟩$

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

$\mathrm{false}$

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

$2$

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

$\mathrm{false}$

$\mathrm{IsMalnormal}\left(\mathrm{SylowSubgroup}\left(5\,G\right)\,G\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(0\,7\,3\right)\right)$

$18341406720$

$\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(1\,8\,2\right)\right)$

$348364800$

$\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(-1\,8\,2\right)\right)$

$394813440$

$\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(-1\,4\,q\right)\right)$

$\mathrm{igcd}\left(2\,q-1\right)\mathrm{igcd}\left(2\,q\right)q^2\left(q^2+1\right)\left(q^2-1\right)$

$\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(1\,4\,q\right)\right)$

$\mathrm{igcd}\left(2\,q-1\right)\mathrm{igcd}\left(2\,q\right)q^2{\left(q^2-1\right)}^2$

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

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

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