Pointwise stabilizer subgroup with canonical generators (named groups; forced PermutationGroup cases)
GroupStabilizer[group, {p1, …, pn}]
GroupStabilizer[group, {p1, …, pn}, f]
GroupStabilizer[AlternatingGroup[6], {1, 5}]
→ PermutationGroup[{Cycles[{{2, 3, 4}}], Cycles[{{3, 4, 6}}]}]GroupStabilizer[AlternatingGroup[10], {10}]
→ PermutationGroup[{Cycles[{{1, 2, 3}}], Cycles[{{2, 3, 4}}], Cycles[{{3, 4, 5}}], Cycles[{{4, 5, 6}}], Cycles[{{5, 6, 7}}], Cycles[{{6, 7, 8}}], Cycles[{{7, 8, 9}}]}]GroupStabilizer[PermutationGroup[{Cycles[{{1, 2, 3}, {4, 5, 6}}]}], {1}]
→ PermutationGroup[{}]