Consider the Leech lattice as a complex lattice over the Eisenstein integers.
Both in Conway ("Sphere packings, lattices and groups") and Wilson ("The Complex Leech Lattice and Maximal Subgroups of the Suzuki Group") it is said that the automorphism group of this lattice is the group (6Sz) which can be generated by its monomial part and a matrix with 3x3 blocks.
From what I've understood, the monomial part consists of automorphisms that are induced by Golay code automorphisms, scalar multiplications and coordinatewise multiplications by elements of the multiplicative Golay code (where $0,1,-1$ are replaced by $1,\omega,\bar{\omega}$ resp.).
Wilson has also said that the matrix \begin{array} {cccccccccccc} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & \omega & \bar{\omega} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & \bar{\omega} & \omega & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & -\bar{\omega} & -\omega & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & -\omega & -\bar{\omega} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \omega & \bar{\omega} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \bar{\omega} & \omega & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -\bar{\omega} & -\omega\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -\omega & -\bar{\omega}\\ \end{array} is an automorphism of the lattice, so this must be the matrix that generates $6Sz$ together with the monomial part.
My problem is that Wilson and Conway label the coordinates differently. I have constructed several elements of the lattice using Conways method and I need to find the matrix as in Wilson.
Conway order: $5,4,3,2,1,0,\infty,6,7,8,9,X$
Wilson order: $\infty,7,6,8,X,2,0,3,4,1,9,5$
How should I understand the matrix of Wilson in Conway's coordinate system?