How do we understand the ${\bf 3}$ of $Q_L({\bf 3}, {\bf 2})_{1/3}$? A left-handed quark doublet of the Standard Model is specified as $Q_L({\bf 3}, {\bf 2})_{1/3}=(u,d)^T$. I have a problem understanding this quark doublet as a triplet of ${\rm SU}(3)$. Any help? I think, we write $(u,d)^T$ as $(u^r,u^b,u^g,d^r,d^b,d^g)^T$ and then the action of SU(3) group on this will be a $6\times 6$ block-diagonal matrix (with $3\times 3$ SU(3) matrices as the blocks) acting on $(u^r,u^b,u^g,d^r,d^b,d^g)^T$. Is this correct?
 A: This has nothing to do with $SU(3)$, the doublet representation is that of the electroweak $SU(2)$. Both $u$ and $d$ are themselves triplets transforming in the fundamental representation $\mathbb{3}$ of $SU(3)$.
Your last sentence sounds correct to me.
A: Yes, you understand right. You represent the quark with $q^i_\alpha$, a simultaneous weak SU(2) doublet and color SU(3) triplet. It is a hypercharge singlet with eigenvalue 1/3 throughout, so that can be skipped.
The color and weak isospin group are in Cartesian product, so the representation you are looking at is a tensor product. So the quark is a 6-vector (spinor) with two distinct i indices and three α indices, as you wrote them down.
A color rotation changes the field by
$$
\delta q^i_\alpha = i \theta_\Gamma \lambda^\Gamma_{\alpha\beta}   q^i_\beta  ~,  
$$
and a weak isorotation by
$$
\delta q^i_\alpha = i \phi_M \tau^M_{ij}   q^j_\alpha  ~,
$$
where Γ, Μ are the color, weak respective adjoint indices.
Representing these as as 6×6 matrices, you may either stick the τ blocks in each entry of the  3D identity for a weak isorotation; or, equivalently, as you implicitly did, the λs in each entry of the 2D identity, for a color rotation. In the first convention, the color rotation is messier; and in the second one, yours, an isorotation is arguably messier, as the Pauli matrices have 3D identities as entries, so the upper three components scramble with the lower three blockwise, e.g.
$$
\delta q  = i \phi_1 \begin{pmatrix}0& {\mathbb I}\\
{\mathbb I} & 0 \end{pmatrix}   ~~  q  .
$$
