Question

Give examples of non-abelian group of orders 12, 16, 30 and 48 .

Short answer

$$\begin{gathered} \left|G\right|=12TakingG=S_3\times {\mathrm{\mathbb{Z}}}_2 \\ \left|G\right|=16TakingG=D_4\times {\mathrm{\mathbb{Z}}}_2 \\ \left|G\right|=30TakingG=S_3\times {\mathrm{\mathbb{Z}}}_5 \\ \left|G\right|=48TakingG=S_3\times {\mathrm{\mathbb{Z}}}_8 \end{gathered}$$

Step-by-step solution

Determine group of orders

Consider the given group orders, and obtain these are non-abelian.

Here using the fact that is$S_6$ and non-abelian therefore its product with any group is non-abelian.

Result

The given group orders are,

$$\begin{gathered} \left|G\right|=12TakingG=S_3\times {\mathrm{\mathbb{Z}}}_2 \\ \left|G\right|=16TakingG=D_4\times {\mathrm{\mathbb{Z}}}_2 \\ \left|G\right|=30TakingG=S_3\times {\mathrm{\mathbb{Z}}}_5 \\ \left|G\right|=48TakingG=S_3\times {\mathrm{\mathbb{Z}}}_8 \end{gathered}$$