Construction

A matrix group is always constructed as a subgroup of the appropriate general linear group, GL(n, R).

• Generators for linear groups: $\mbox{\rm GL}(n,q)$, $\mbox{\rm SL}(n,q)$
• Generators for symplectic groups: $\mbox{\rm Sp}(n,q)$
• Generators for unitary groups: $\mbox{\rm GU}(n,q)$, $\mbox{\rm U}(n,q)$
• Generators for orthogonal groups: $\mbox{\rm GO}(2n+1,q)$, $\mbox{\rm SO}(2n+1,q)$, $\Omega(2n+1,q)$, $\mbox{\rm GO}^{+}(2n,q)$, $\mbox{\rm SO}^{+}(2n,q)$, $\Omega^{+}(2n,q)$, $\mbox{\rm GO}^{-}(2n,q)$, $\mbox{\rm SO}^{-}(2n,q)$, $\Omega^{-}(2n,q)$
• Generators for all exceptional families of groups of Lie type except E(8).
• Direct product, tensor wreath product, tensor power, exterior square
• Construction of semi-linear groups
• Group obtained by applying a monomorphism $\phi: R \rightarrow S$ to the matrix coefficients.
• Group obtained by restricting the matrix coefficients to a subring of R.