Generators of $O_8^+(2):\mathbb Z_2$

Question

I am trying to find out what the generators of $O_8^+(2):\mathbb Z_2$ are, and they are given in the ATLAS of Finite Group Representations. However I do not understand the explanation given:

Standard generators of O8+(2):2 are c, d where c is in class 2F, d is in class 10BC and cd has order 18. Alternatively: c is in class 2F, d has order 10, cd has order 18 and cdd has order 30.

What does class 2F and 10BC mean? Under "presentation" section, stuff are crossed out.

So the question is what are the generators and what are the complete set of relations they must satisfy to generate this group.

(Bonus question, is this group related to the Weyl group of $E_8$? I've read that the orientation preserving subgroup of $W(E_8)$ is a $\mathbb Z_2$ extension of $O_8^+(2)$.)

Answer 1

The following Magma script uses the alternative description of $c$ and $d$ to find them. From the ATLAS, we know that $c$ lies outside of $O_8^+(2)$ and has centralizer in $O_8^+(2)$ of order 1451520. That's enough to locate it from a list of conjugacy class representatives.

> G:=SOPlus(8,2);
> C:=Classes(G);
> C2 := [c : c in C | 
         c[1] eq 2 and not c[3] in DerivedGroup(G)];
> #C2;
2
> [#Centraliser(G,c[3]) : c in C2];
[ 2903040, 9216 ]
> c := C2[1][3]; //in class 2F
> repeat
  repeat> repeat d:=Random(G); od := Order(d);
          until od mod 10 eq 0;
  repeat> d := d^(od div 10);  //now d has order 10
  repeat> until Order(c*d) eq 18 and Order(c*d*d) eq 30;
> Order(c), Order(d), Order(c*d), Order(c*d*d);
2 10 18 30
> sub< G | c,d> eq G;
true
> c,d;
[1 0 1 0 0 1 1 1]
[0 1 1 0 0 1 1 1]
[0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]

[1 1 0 0 1 0 1 1]
[1 0 0 1 1 1 1 1]
[1 1 1 0 0 0 1 1]
[0 1 0 0 0 0 0 1]
[0 0 0 0 1 0 1 1]
[1 1 0 0 1 1 1 1]
[1 1 0 0 0 0 1 1]
[0 1 1 1 0 0 0 1]