Determine the set of the conjugators between two affine unimodular matrix groups, whose linear parts are the same finite group belonging to the subgroup of GLNZ.

[Hongyi Zhao <hongyi.zhao@gmail.com>]

Hi here,

I have two affine unimodular matrix groups whose generators are as follows:

gens2=[
[  -1, 0, 0, 0 ; 0, 1, 0, 1; 0, 0, 1, 0; 0, 0, 0, 1 ],
[ -1, 0, 0, 0; 0, 1, 0, -1/2; 0, 0, -1, 0; 0, 0, 0, 1 ],
[ -1, 0, 0, 0; 0, -1, 0, 1/2; 0, 0, 1, 0; 0, 0, 0, 1 ],
[ 1, 0, 0, 1/2; 0, 1, 0, 1/2; 0, 0, 1, 1/2; 0, 0, 0, 1 ] ]

gens3=[
[ 1, 0, 0, 0; -1, -1, 0, 0; 0, 0, 1, -1/2; 0, 0, 0, 1 ],
[ -1, 0, 0, 0; 0, -1, 0, 1/2; 1, 0, 1, 0; 0, 0, 0, 1 ],
[ 1, 0, 0, 0; -1, -1, 0, 0; -1, 0, -1, -1/2; 0, 0, 0, 1 ],
[ 1, 0, 0, -1; 0, 1, 0, 1; 0, 0, 1, 1; 0, 0, 0, 1 ] ]

# Sorry here once more, I really don't know what's corresponding
syntax for this in PARI/GP:
grp2=Group(gens2)
grp3=Group(gens3)

Here, the 3-by-3 matrices corresponding to linear parts of the above
two set of generators form the same finite group G, which is a
subgroup in GLNZ.

I want to determine the set of the conjugators between these two groups, a.k.a.,

conjs = { c | grp2 ^ c = grp3  }

Where, the linear part of c is an element of the normalizer of G, and
the translation part of c is a rational vector which has the following
form:

[ x, y, z, 1 ]

If there's no such conjugator, how can I make the decision quickly?

Regards,
Zhao
-- 
Assoc. Prof. Hongsheng Zhao <hongyi.zhao@gmail.com>
Theory and Simulation of Materials
Hebei Vocational University of Technology and Engineering
No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province