Group generated by abelian normal subgroups

Symbol-free definition
A group is said to be generated by abelian normal subgroups if there exists a collection of abelian normal subgroups which together generate the group.

Examples
The dihedral group of size eight and the quaternion group are examples of non-Abelian groups generated by Abelian normal subgroups. While the former is generated by a cyclic normal subgroup of order 4 and a Klein four-group, the latter is generated by two cyclic normal subgroups.

Weaker properties

 * nilpotent group (for finite groups):

Facts

 * Finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class