Normal subgroups of signed symmetric groups

For a symmetric group $$S_n$$, we have known clearly the proper nontrivial normal subgroups of it: if n=3 or $$\geq 5$$, $$S_n$$ has only one normal subgroup; if n=4, $$S_n$$ has two normal subgroups; otherwise, $$S_n$$ has no normal subgroups.

As a generalization of $$S_n$$ is the signed symmetric group:https://groupprops.subwiki.org/wiki/Signed_symmetric_group. Could someone tell me what are the normal subgroups of signed symmetric groups? (I think they should be very similar like $$S_n$$) Thank you.

• See A. Kerber, Representations of Permutation Groups I. Springer-Verlag (LNM 240), (1971). The signed symmetric group is a special case of the generalized symmetric group, which is discussed in Kerber's book. – Dietrich Burde Nov 19 '19 at 9:12
• @ Dietrich Burde Thank you for your help. – Xiaosong Peng Nov 19 '19 at 9:42
• The number of normal subgroups of $S_n$ for $n=0,1,2,3,4,\ge 5$ is $1,1,2,3,4,3$. For these values the number of proper normal subgroups is $0,0,1,2,3,2$; the number of nontrivial proper normal subgroups is $0,0,0,1,2,1$. – YCor Nov 19 '19 at 22:00

In brief, in ATLAS notation, for $$n \ge 5$$ there are $$9$$ normal subgroups of the signed permutation group $$2^n\!:\!S_n$$.
These have structures $$1$$, $$2$$, $$2^{n-1}$$, $$2^n$$, $$2^{n-1}\!\!:\!A_{n}$$, $$2^n\!:\!A_n$$, $$2^{n-1}\!\!:\!S_n$$ (two subgroups) and $$2^n\!:\!S_n$$.
Note that the normal subgroup $$2^{n-1}$$ consists of diagonal matrices of determinant $$1$$ in the standard linear representation. Also $$A_n$$ acts irreducibly on $$2^{n-1}$$ when $$n$$ is odd, but when $$n$$ is even, the subgroup $$2$$ is contained in the subgroup $$2^{n-1}$$, with irreducible action of $$A_n$$ on the quotient of order $$2^{n-2}$$.
There are 11 normal subgroups when $$n=4$$, 9 when $$n=3$$, and 6 when $$n=2$$.