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\noindent A finite group $G$ is called $n$-decomposable if it is
non-simple and each of its non-trivial proper normal subgroups is
a union of $n$ distinct conjugacy classes. In this paper, we
investigate the structure of non-solvable non-perfect finite group
$G$ when $G$ is 5- or 6-decomposable. We prove that $G$ is
5-decomposable if and only if $G$ is isomorphic with $Z_5\times
A_5$, $A_6.2_3$ or Aut(PSL (2,q)) for $q = 7,8$. Also, $G$ is
6-decomposable if and only if $G$ is isomorphic with $S_6$ or
$A_6.2_2$. Here, $A_6. 2_2$ and $A_6. 2_3$ are non-isomorphic
split extensions of the alternating group $A_6$, in the small
group library of GAP [SCHONERT,M. et al.: GAP, \emph
{Groups,Algorithms and Programming}. Lehrstuhl f\"{u}r Mathematik,
RWTH, Aachen, 1992].
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