Twin sextic algebra (reviewed)

If $A$ is a separable algebra of degree 6 over a field $F$ (so $\dim_F A=6$), then it has a twin algebra $B$ that is also a separable algebra of degree 6 over $F$; the $F$-algebras $A$ and $B$ are related by the non-trivial outer automorphism of $S_6$.

The symmetric group $S_n$ has a non-trivial outer automorphism if and only if $n=6$, in which case it is unique up to inner automorphisms. This automorphism interchanges $S_5\leq S_6$ with $\PGL_2(5)\leq S_6$. In terms of Galois theory, this inclusion gives an absolute resolvent of degree $[G:\PGL_2(5)]=6$.

A common situation is when $A$ is a field but $B$ is not. Then the nature of the factorization of $B$ assists in the study of $A$.