Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$, 
the group of invertible upper triangular $n\times n$ matrices. 
I know that if $\rho : G\rightarrow T(n,k)$ is faithful 
(i.e. into) then $G$ is resoluble, we have also theorems for the 
converse of the form (Lie-Kochin)



> 
> $G\mathrm{\ is\ resoluble\ +\ conditions}(G,k)$ $\Longrightarrow$   
> 
>  $\mathrm{it\ exists}$ $n$ $\mathrm{and\ a\ faithful\ representation}$ $\rho : G\rightarrow T(n,k)$
> 
> 
> 


What is known if "*finite*" is one of the conditions ? (i.e. for finite resoluble 
groups)



Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
A finite group of upper triangular matrices has a nilpotent derived group, so a solvable group whose derived group is not nilpotent can't have a faithful representation by upper triangular matrices. One such group is ${\rm GL}(2,3)$ whose derived group is ${\rm SL}(2,3)$, which is certainly not nilpotent (actually $S\_{4}$ also works and is a more or less equivalent (but simpler) example).


Later edit: When $k$ has finite characteristic $p >0$, we in fact need $[G,G]$ to be a $p$-group to have any chance of a faithful representation by upper triangular matrices. It is the case that when $G$ is finite and $[G,G]$ is a $p$-group, then $G$ does have a faithful representation by upper triangular matrices over some finite field $k$ of characteristic $p$. For $G/O\_{p}(G)$ is then a finite Abelian $p^{\prime}$-group, so all absolutely irreducible representations of $G$ in characteristic $p$ are $1$-dimensional. It follows that there is a finite field $k$ of characteristic $p$ such that the regular representation of $G$ over $k$ (which is certainly faithful) is equivalent as a matrix representation to a representation of $G$ by upper triangular matrices over $k$. 


Even later edit: I suppose I should remark for completeness that when $k$ has characteristic $0$, the only finite solvable groups which have a faithful representation by upper triangular matrices with entries in $k$ are Abelian. For if $G$ is such a group, then $[G,G]$ consists of unipotent matrices ( in the chosen faithful representation), and the identity matrix is the only unipotent matrix of finite order in ${\rm GL}(n,k)$ when $k$ has characteristic $0$.


