Dear Forum,

I have tried to use GO(-1,6,2) but encountered

gap> matGroup := GeneralOrthogonalGroup(-1,6,2);
gap> InvariantQuadraticForm(matGroup);
rec(
matrix := [ <an immutable GF2 vector of length 6>, <an immutable GF2
vector of length 6>, <an immutable GF2 vector of length 6>,
<an immutable GF2 vector of length 6>,
<an immutable GF2 vector of length 6>,
<an immutable GF2 vector of length 6> ] )

However when I choose q=3 or 5 I get the quadratic form which gives the
group, and then I can do all kinds of computations. How do I generate the
group when q is even?

There is no problem in fact, although you might be confused by the unusual
way the matrix is presented here. In fact it's quite a normal matrix:

gap> matGroup := GeneralOrthogonalGroup(-1,6,2);            
GO(-1,6,2)
gap> Print(InvariantQuadraticForm(matGroup).matrix);
[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]gap> 

while
gap> InvariantQuadraticForm(matGroup).matrix;
[ <an immutable GF2 vector of length 6>, <an immutable GF2 vector of length
6>, <an immutable GF2 vector of length 6>,
<an immutable GF2 vector of length 6>, <an immutable GF2 vector of length
6>, <an immutable GF2 vector of length 6> ]

HTH,
Dmitrii

PS. One could argue that such a external representation for
immutable GF(2) vectors is not very user-friendly.