I encounter a problem in my research: Let $L$ be the maximal pro-2 extension unramified outside $2, 3, \infty$ over $Q$, I hope I could know some information about the Galois group $Gal(L/Q)$. However, based on my poor knowledge of Galois theory, I have to admit that it's a little bit hard to me. Is there any people who could provide some information about that group? Any related information will be welcome.
Thanks very much!
The answer to your question (in a slightly more general situation) is : Let $\mathbf Q_S/\mathbf Q$ be the maximal pro-2-extension unramified outside $S= ${$2, q, \infty$}, where $q$ is a prime $\equiv \pm 3$ mod $8$, and denote $G_S = Gal(\mathbf Q_S/\mathbf Q)$. Then $G_S$ can be described by generators and relations as follows : $G_S$ is the quotient of the free pro-2-group with generators $s_q, t_q, t_\infty$ modulo the normal subgroup generated by $t_q^{q-1}[t_q^{-1}, s_q^{-1}]$ and $t_\infty^2$. See the example 11.18 at the end of chap. 11 of [K]. How is this proved ? Because of your "poor knowledge of Galois theory" (*) , and consequently your even poorer knowledge of Galois cohomology, I can only give a very sketchy idea of the enormous machinery at work (see the references below).
Given a number field $K$, a rational prime $p$, and a finite set $S$ of primes of $K$ containing the set $S_p$ of primes of $K$ above $p$ as well as the set $S_{\infty}$ of infinite primes, the goal is the determination of $G_S = G_S (K)=Gal(K_S/K)$ where $K_S$ is the maximal pro-$p$-extension of $K$ unramified outside $S$. One must distinguish fundamentally between two cases because, by the definition of ramification, the following primes cannot ramify in a $p$-extension : the finite primes $P$ s.t. $N(P)\neq 1\pmod p$, the complex primes, and the real primes if $p$ is odd.
In the first case, $p$ is odd and $K$ is totally imaginary. In view of the "global-local" principle in ANT, one first describes by generators and relations the Galois group $G_v$ of the maximal pro-$p$-extensions of a local field $K_v$ ( = completion of $K$ at a prime $v\in S$). If $v\notin S_p$ (tame ramification), $G_v$ is described by a single simple relation. When $v\in S_p$ (wild ramification) the problem is solved by two deep results : if $K_v$ does not contain a primitive $p$-th root of unity, $G_v$ is pro-$p$-free (Shafarevitch), and if if $K_v$ contains such a root, $G_v$ is described by a single complicated relation (Demushkin). Back to the group $G_S$ : intuitively, its relations must include the local ones, but unfortunately, there are also truly global ones which, loosely speaking, are attached to the $\mathbf Z_p$-torsion subgroup of the abelianization of $G_S$. See [K], chap. 11, or [NSW], chap. 10 . The second case, which includes your question, is technically more complicated (as always when the prime $2$ is involved, think e.g. of the distinction between the class group and the narrow class group). See details in [NSW], chap. 10, §6. Of course, your question can be entirely solved only thanks to the simplications brought by the fact that the base field is $\mathbf Q$ .
[K] H. Koch : "Galoissche Theorie der $p$-Erxeiterungen"
[NSW] Neukirch-Schmidt-Wingberg : "Galois cohomology of number fields"
(*) NB. But then, how did you get interested in this specific question ?