The affine special linear group $\mathrm{ASL}\left(n\,q\right)$ is the semi-direct product of the special linear group $SL\left(n\,q\right)$ with the natural module of dimension $n$ over the field with $q$ elements. It is also called the special affine group, and is sometimes denoted by $\mathrm{SA}\left(n\,q\right)$.

The AffineSpecialLinearGroup command produces a permutation group isomorphic to the group $\mathrm{ASL}\left(n\,q\right)$.

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

$G≔\mathrm{AffineSpecialLinearGroup}\left(1\,2\right)$

$G≔\mathbf{ASL}\left(1\,2\right)$

$G≔\mathrm{ASL}\left(1\,2\right)$

$G≔\mathbf{ASL}\left(1\,2\right)$

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

$2$

$Q≔\mathrm{select}\left(\mathrm{type}\,\left[\mathrm{seq}\right]\left(2..100\right)\,'\mathrm{primepower}'\right)\:$

$G≔\mathrm{map2}\left(\mathrm{ASL}\,1\,Q\right)\:$

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

$\left[2\,3\,4\,5\,7\,8\,9\,11\,13\,16\,17\,19\,23\,25\,27\,29\,31\,32\,37\,41\,43\,47\,49\,53\,59\,61\,64\,67\,71\,73\,79\,81\,83\,89\,97\right]$

$\mathrm{andmap}\left(\mathrm{IsElementary}\,G\right)$

$\mathrm{true}$

$G≔\mathrm{ASL}\left(2\,2\right)$

$G≔\mathbf{ASL}\left(2\,2\right)$

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

$\mathrm{true}$

$G≔\mathrm{ASL}\left(3\,3\right)$

$G≔\mathbf{ASL}\left(3\,3\right)$

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

$\mathrm{true}$

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

$2$

$S≔\mathrm{Socle}\left(G\right)$

$S≔⟨\left(1\,19\,10\right)\left(2\,20\,11\right)\left(3\,21\,12\right)\left(4\,22\,13\right)\left(5\,23\,14\right)\left(6\,24\,15\right)\left(7\,25\,16\right)\left(8\,26\,17\right)\left(9\,27\,18\right)\,\left(1\,3\,2\right)\left(4\,6\,5\right)\left(7\,9\,8\right)\left(10\,12\,11\right)\left(13\,15\,14\right)\left(16\,18\,17\right)\left(19\,21\,20\right)\left(22\,24\,23\right)\left(25\,27\,26\right)\,\left(1\,4\,7\right)\left(2\,5\,8\right)\left(3\,6\,9\right)\left(10\,13\,16\right)\left(11\,14\,17\right)\left(12\,15\,18\right)\left(19\,22\,25\right)\left(20\,23\,26\right)\left(21\,24\,27\right)⟩$

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

$27$

$\mathrm{IsElementary}\left(S\right)$

$\mathrm{true}$

$\mathrm{IsRegular}\left(S\right)$

$\mathrm{true}$