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 $\mathrm{ASL}\left(n\,q\right)$.

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

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

$G≔\mathbf{ASL}\left(1\,2\right)$

$G≔\mathrm{ASL}\left(1\,2\right)$

$G≔\mathbf{ASL}\left(1\,2\right)$

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

$2$

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

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

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

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

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

$\mathrm{true}$

$G≔\mathrm{ASL}\left(2\,2\right)$

$G≔\mathbf{ASL}\left(2\,2\right)$

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

$\mathrm{true}$

$G≔\mathrm{ASL}\left(3\,3\right)$

$G≔\mathbf{ASL}\left(3\,3\right)$

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

$\mathrm{true}$

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

$2$

$S≔\mathrm{Socle}\left(G\right)$

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

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

$27$

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

$\mathrm{true}$

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

$\mathrm{true}$