The ReducedDegreePermGroup( G ) command returns a permutation group isomorphic (as an abstract group) with possibly smaller degree, if one can be found.

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

$G≔\mathrm{SmallGroup}\left(48\,15\right)$

$G≔⟨\left(1\,2\right)\left(3\,15\right)\left(4\,11\right)\left(5\,10\right)\left(6\,12\right)\left(7\,14\right)\left(8\,13\right)\left(9\,16\right)\left(17\,28\right)\left(18\,27\right)\left(19\,38\right)\left(20\,37\right)\left(21\,34\right)\left(22\,33\right)\left(23\,32\right)\left(24\,31\right)\left(25\,36\right)\left(26\,35\right)\left(29\,40\right)\left(30\,39\right)\left(41\,48\right)\left(42\,47\right)\left(43\,46\right)\left(44\,45\right)\,\left(1\,3\right)\left(2\,9\right)\left(4\,17\right)\left(5\,16\right)\left(6\,18\right)\left(7\,19\right)\left(8\,20\right)\left(10\,27\right)\left(11\,15\right)\left(12\,28\right)\left(13\,29\right)\left(14\,30\right)\left(21\,41\right)\left(22\,42\right)\left(23\,39\right)\left(24\,40\right)\left(25\,43\right)\left(26\,44\right)\left(31\,45\right)\left(32\,46\right)\left(33\,37\right)\left(34\,38\right)\left(35\,47\right)\left(36\,48\right)\,\left(1\,4\,6\,5\right)\left(2\,10\,12\,11\right)\left(3\,16\,18\,17\right)\left(7\,21\,25\,23\right)\left(8\,22\,26\,24\right)\left(9\,15\,28\,27\right)\left(13\,31\,35\,33\right)\left(14\,32\,36\,34\right)\left(19\,39\,43\,41\right)\left(20\,40\,44\,42\right)\left(29\,37\,47\,45\right)\left(30\,38\,48\,46\right)\,\left(1\,6\right)\left(2\,12\right)\left(3\,18\right)\left(4\,5\right)\left(7\,25\right)\left(8\,26\right)\left(9\,28\right)\left(10\,11\right)\left(13\,35\right)\left(14\,36\right)\left(15\,27\right)\left(16\,17\right)\left(19\,43\right)\left(20\,44\right)\left(21\,23\right)\left(22\,24\right)\left(29\,47\right)\left(30\,48\right)\left(31\,33\right)\left(32\,34\right)\left(37\,45\right)\left(38\,46\right)\left(39\,41\right)\left(40\,42\right)\,\left(1\,7\,8\right)\left(2\,13\,14\right)\left(3\,19\,20\right)\left(4\,21\,22\right)\left(5\,23\,24\right)\left(6\,25\,26\right)\left(9\,29\,30\right)\left(10\,31\,32\right)\left(11\,33\,34\right)\left(12\,35\,36\right)\left(15\,37\,38\right)\left(16\,39\,40\right)\left(17\,41\,42\right)\left(18\,43\,44\right)\left(27\,45\,46\right)\left(28\,47\,48\right)⟩$

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

$48$

$R≔\mathrm{ReducedDegreePermGroup}\left(G\right)$

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

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

$24$

$G≔\mathrm{CyclicGroup}\left(9\right)$

$G≔C_9$

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

$9$

$R≔\mathrm{ReducedDegreePermGroup}\left(G\right)$

$R≔C_9$

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

$9$

$G≔\mathrm{MathieuGroup}\left(11\,'\mathrm{form}'=''fpgroup''\right)$

$G≔⟨a\,b\mid a^2\,b^4\,a{b}^{2}a{b}^{2}a{b}^{2}a{b}^{2}a{b}^{2}a{b}^2\,ababa{b}^{-1}aba{b}^{2}a{b}^{-1}aba{b}^{-1}a{b}^{-1}\,ababababababababababab⟩$

$P≔\mathrm{PermutationGroup}\left(G\right)\:$

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

$7920$

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

$\mathrm{true}$

$R≔\mathrm{ReducedDegreePermGroup}\left(P\right)$

$R≔⟨\left(1\,2\right)\left(4\,5\right)\left(7\,9\right)\left(10\,11\right)\,\left(1\,2\,4\,3\right)\left(5\,6\,8\,7\right)\left(9\,10\right)\left(11\,12\right)⟩$

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

$12$