The problem is to find a presentation of the regular wreath product of $C_4$ and $D_6$ (the dihedral group of order 6).
Note: By definition the wreath product is a semidirect product $(C_4)^6 \rtimes D_6$ realised by a specific homomorphism $ \theta$, one should inculude the relations of the free product $(C_4)^6 * D_6$:
{$a^2=...=f^2=1, [a,b]=[a,c]=...=[e,f]=1, x^6=y^2=1, xy=yx^-1$} besides one needs adding the conjugates $a^x,a^y, b^x,b^y...$.
My query: What tuple should $a$ be taken as in order to compute $a^x=\theta_x (a)$? For example given $a$ is the generator of the first (left-most) copy of $C_4$ in $(C_4)^6$ then is $\theta_x (a)=\theta_x((1,0,0,0,0,0))$?