Thanks for any comments or help.
What is the list of faithful permutation representations of $S_k$ of degree at most $n=2k$ for $k=3,4,5,6$? Is it possible to find its centralizer in each case?
2026-04-02 09:16:17.1775121377
Permutation representations of symmetric group of small order
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This is easy to do on a computer. For example in magma http://magma.maths.usyd.edu.au/calc/
The following code:
for k in [3..6] do X:=Sym(k); for n in [k..2*k] do G:=Sym(n); for sub in Subgroups(G:OrderEqual:=Factorial(k)) do H:=sub`subgroup; if IsIsomorphic(H,X) then [k,n]; Order(Centraliser(G,H)); end if; end for; end for; end for;
will give you all the representations (up to conjugacy in Sym(n)) and the order of the centraliser. It runs in about 10 seconds.
(For example, there are 10 conjugacy classes of Sym(6) in Sym(12), with centralisers of order between 1 and 720.) If you want more specific information, you can modify the program as needed.