Frattini-in-center group

Definition
A Frattini-in-center group is a group satisfying the following equivalent conditions:


 * Its defining ingredient::inner automorphism group (i.e., the quotient group by its center) is Abelian and Frattini-free.
 * Its defining ingredient::commutator subgroup is contained in its defining ingredient::Frattini subgroup, which in turn is contained in its defining ingredient::center.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Special group
 * Weaker than::Extraspecial group
 * Weaker than::Group of Frattini class two

Weaker properties

 * Stronger than::Group of nilpotency class two

Facts
A group of prime power order is Frattini-in-center if and only if it satisfies the following equivalent conditions:


 * It is a critical subgroup of itself.
 * It can be realized as a critical subgroup of some group of prime power order.