Consider the set $A_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\cdots, n-1,n\}$ where $n \in \mathbb{N}$. Let G be a subgroup of the symmetric group $S_{A_n}$ defined as follow: $G=\{\sigma \in S_{A_n}| \forall x \in A_n, \sigma(-x)= -\sigma(x)\}$.
I think $G$ can be generated by the set formed by the transpositions $(i , -i), 1\le i \le n$ and $(1,2)(-1,-2), (1,2, \cdots, n)(-1,-2, \cdots, -n)$ but I don't know if it is an irredundant set of generators or even a minimal subset of generators.
What is an example of irredundant set of generators of $G$?
What is the size of the smallest irredundant set of generators of $G$?
G is also known as the signed symmetric group, denoted $B_n$.