Can I have some examples of finite noncommutative connected group schemes over a field $k$?
I would like also to see some nontrivial torsors over a $k$scheme $X$ under such group schemes. Thanks.
Can I have some examples of finite noncommutative connected group schemes over a field $k$? I would like also to see some nontrivial torsors over a $k$scheme $X$ under such group schemes. Thanks. 


If $\mathrm{char}(k)=p>0$ and $G$ is a $k$group scheme of finite type, the kernel of the relative frobenius $F_{G/k}:G\to G^{(p)}$ is a finite connected $k$group scheme. It has the same Lie algebra as $G$, and in particular it is noncommutative if the Lie algebra is nonabelian, e.g. for $G=GL_{n,k}$, $n\geq2$. Remark: the example given by anon as a comment to the question may be seen as the special case where $G$ is the natural semidirect product $\mathbb{G}_a \rtimes \mathbb{G}_m$, i.e. the group of affine automorphisms of the affine line. 

