# Transposition as generating $S_n$.

As far as I know the group of symmetries of the euclidean n-cube is generated by reversions ${r_i}$ and transpositions ${s_i}$ for i=1,...,n-1. Transpositions generate the subgroup of permutations of coordinates, which is isomorphic to $S_n$, the symmetric group, acting by swapping two coordinates and leaving the others unchanged:

$s_i(a_1,...,a_n)=(a_1,...,a_{i-1},a_{i+1},a_i,a_{i+2},...,a_n)$

I need to know if there is a known set of transpositions which considers also (perhaps starting from n=3) the transposition

$s_n(a_1,...,a_n)=(a_n,a_2,...,a_{n-1},a_1)$

and which group does it generate.

This is not a transposition, but in any case it is a permutation and hence is an element of $S_n$. The group of symmetries of a hypercube may be combinatorially realized as the hyperoctahedral group, or the group of signed permutations. These are permutations $f$ of $\{-n,-n+1,\ldots,-1,1,2,\ldots,n\}$ such that $f(-i)=-f(i)$ for all $i$ and are usually identified by the sequence of values $(f(1),f(2),\ldots,f(n))$. This group is generated as a Coxeter group by the adjacent transpositions together with the element $(-1,2,3,\ldots,n)$.