Construction of a triple cover of $A_6$ in "Finite Simple Groups" by Wilson

Question

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is following:

In addition to the above ... i.e. the map..

Could someone clarify this?

Given vectors $(0,0,1,1,1,1), \ (0,1,0,1,\omega, \omega^2)$, if I take the multiples of $\omega $ and $\omega^2$ and map them under $S_4$ as said in the first paragraph, I understand that one gets a set of 45 vectors.

Now, my questions are:

1) What does monomial elements mean? There is given an example but what are others?

2) In the third paragraph, he says "This group....". What is he referring to? How do we get $G$?

ADDED: Why does $G$ induce all even permutations?

Thanks!

Answer 1

1. "Monomial element" refers to a monomial matrix; a matrix with exactly one non-zero entry in each row and column (also known as a generalised permutation matrix, cf. Wikipedia). But in the paragraph you're reading it is enough to take the bracketed sentence to be the definition (i.e. a monomial matrix is a product of a permutation matrix with a diagonal matrix).

2. "This group" refers to the group generated by all the symmetries considered so far, i.e. the coordinate permutations $S_{4}$ and the "other monomial elements".