$\overrightarrow{a}\cdot \overrightarrow{d}=\overrightarrow{b}\cdot \overrightarrow{d}=\overrightarrow{c}\cdot \overrightarrow{d}=0$ (Given)

$$\begin{gathered} \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=\left|\overrightarrow{d}\right|=1\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ } \\ \mathrm{ }\overrightarrow{a}\cdot \overrightarrow{b}=\frac{\sqrt{3}}{2} \end{gathered}$$
$\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \left(\overrightarrow{c}\times \overrightarrow{d}\right)=\frac{3}{26}\widehat{i}-\frac{2}{13}\widehat{j}+\frac{6}{13}\widehat{k}$

$\Rightarrow \left(\left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \overrightarrow{d}\right)\overrightarrow{c}-\left(\left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \overrightarrow{c}\right)\overrightarrow{d}=\frac{3}{26}\hat{i}-\frac{2}{13}\hat{j}+\frac{6}{13}\hat{k}$   ...$\left(1\right)$

$\because \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are coplanar $\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \overrightarrow{c}=0$

$\because \overrightarrow{d}$ is perpendicular to $\overrightarrow{a},\overrightarrow{b}\Rightarrow \overrightarrow{d}$ must be parallel to $\overrightarrow{a}\times \overrightarrow{b}$

So, the angle between $\overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{b}$ is either $0^{\mathrm{o}}$ or $180^{\mathrm{o}}$

From $\left(1\right)$, we get,

$\pm \left|\overrightarrow{a}\times \overrightarrow{b}\right|\left|\overrightarrow{d}\right|\overrightarrow{c}=\frac{2}{26}\widehat{i}-\frac{2}{13}\widehat{j}+\frac{6}{13}\widehat{k}$

$\Rightarrow \pm \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin {30}^{\mathrm{o}}\left|\overrightarrow{d}\right|\overrightarrow{c}=\frac{3}{26}\hat{i}-\frac{2}{13}\hat{j}+\frac{6}{13}\hat{k}$

$$\begin{gathered} \Rightarrow \pm \frac{\overrightarrow{c}}{2}=\frac{3}{26}\hat{i}-\frac{2}{13}\hat{j}+\frac{6}{13}\hat{k} \\ \Rightarrow \overrightarrow{c}=\pm \frac{\left(3\hat{i}-4\hat{j}+12\hat{k}\right)}{13} \end{gathered}$$