For every prime number
p
≥
3
p\geq 3
and every integer
m
≥
1
m\geq 1
, we prove the existence of a continuous Galois representation
ρ
:
G
Q
→
G
l
m
(
Z
p
)
\rho : G_\mathbb {Q} \rightarrow Gl_m(\mathbb {Z}_p)
which has open image and is unramified outside
{
p
,
∞
}
\{p,\infty \}
if
p
≡
3
p\equiv 3
mod
4
4
and is unramified outside
{
2
,
p
,
∞
}
\{2,p,\infty \}
if
p
≡
1
p \equiv 1
mod
4
4
. We also revisit the question of the lifting of residual Galois representations in terms of embedding problems; that allows us to produce Galois representations with open image in the group of upper triangular matrices with diagonal entries equal to
1
1
, unramified outside
{
p
,
∞
}
\{p,\infty \}
, for
m
m
“small” comparing to
p
p
.