Type of groups (reviewed)

For a finite group $G$, we write Type - length as follows

• If $G$ is abelian then we write "Abelian - $n$", where $n$ is the number of cyclic groups in the unique cyclic decomposition of $G$ as $C_{d_1} \times C_{d_2}\times \cdots \times C_{d_n}$ with $d_i\mid d_{i+1}$ and $d_i>0$.
• If $G$ is nilpotent but not abelian then we write "Nilpotent - $n$", where $n$ is the nilpotency class of $G$.
• If $G$ is solvable but not nilpotent then we write "Solvable - $n$", where $n$ is the derived length of $G$.
• If $G$ is not solvable then we write "Non-solvable - $n$", where $n$ is the number of Jordan factors of $G$.