  The dominant theme of this thesis is the construction of matrix
representations of finite solvable groups using a suitable system of
generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots
p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series:
$\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that
$G_{i}/G_{i-1}$ is isomorphic to a cyclic group of order $p_{i}$, $i =
1,2,\dots,n$. Associated with this series, there exists a system of generators
consisting $n$ elements $x_{1}, x_{2}, \dots, x_{n}$ (say), such that $G_{i} =
\langle x_{1}, x_{2}, \dots, x_{i} \rangle$, $i = 1,2,\dots,n$, which is called
a "long system of generators". In terms of this system of generators and
conjugacy class sum of $x_{i}$ in $G_{i}$, $i = 1,2, \dots, n$, we present an
algorithm for constructing the irreducible matrix representations of $G$ over
$\mathbb{C}$ within the group algebra $\mathbb{C}[G]$. This algorithmic
construction needs the knowledge of primitive central idempotents, a well
defined set of primitive (not necessarily central) idempotents and the
"diagonal subalgebra" of $\mathbb{C}[G]$. In terms of this system of
generators, we give simple expressions for the primitive central idempotents, a
well defined system of primitive (not necessarily central) idempotents and a
convenient set of generators of the "diagonal subalgebra" of $\mathbb{C}[G]$.
  For a finite abelian group, we present an algorithm for constructing the
inequivalent irreducible matrix representations over a field of characteristic
$0$ or prime to the order of the group and a systematic way of computing the
primitive central idempotents of the group algebra. Besides that, we give
simple expressions of the primitive central idempotents of the rational group
algebra of a finite abelian group using a "long presentation" and it's
Wedderburn decomposition.
