Action of lattice automorphism group on discriminant group
I have a lattice $L$ with automorphism group $G=Aut(L)$. The action of $G$ on $L$ should induce an action on the discriminant group $D={\tilde L}/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $\tilde L$ is the weight lattice of $A_2$ and the discriminant group ${\tilde L}/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.
Please provide a concrete situation, where the involved objects are still "complicated" (at least not trivial). Two examples - one in a smaller dimension, one in a much bigger one - would be enough to get started for a potential helper...
I added additional details and a smaller dimensional example of the general question. I don't think entering the 20x20 gram matrix of the lattice would be particularly helpful to anyone. p.s. sorry for bad LaTeX but for some reason $L^*$ didn't work properly.
@jah -- actually, providing an actual explicit example, that people can copy-paste into a Sage session to get started working on your question, is usually an enormous help.