# Series-equivalent characteristic central subgroups may be distinct

From Groupprops

## Statement

It is possible to have a finite group (in fact, even a group of prime power order) and characteristic subgroups and of such that:

- and are Series-equivalent subgroups (?) of : and are isomorphic groups and the quotient groups and are also isomorphic groups.
- and are both Central subgroup (?)s of , i.e., they are both contained in the center of . In particular, both are Characteristic central subgroup (?)s of .
- is
*not*equal to , i.e., and are distinct subgroups.

## Related facts

See series-equivalent not implies automorphic#Related facts.

## Proof

The smallest example is where is SmallGroup(32,28), and are both characteristic central subgroups of order 2, and and are both isomorphic to direct product of D8 and Z2.