# Conditions for a solvable group to have a non-trivial center

I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for me if I knew that the center was non-trivial. Here is what I know about the group:

• $$G$$ is solvable.
• the derived subgroup $$G'$$ is a $$p$$-group.
• $$G''$$ is the unique minimal normal subgroup in G.
• $$G$$ has a faithful irreducible character.

Of course, a lot of information can be deduced from the above. But in particular, I want to know if $$G$$ has a non-trivial center. This may not be deduced from the above information, but maybe if some additional condition is satisfied?

• Any finite group with a unique minimal normal subgroup must have a faithful irreducible character, so your final condition is redundant. – Derek Holt Oct 25 '18 at 8:27
• Your group has derived length $3,$ so I am curious to know how this fits with the context you state for the question ( in other words, how would it help to have a non-trivial center)? – Geoff Robinson Oct 25 '18 at 9:17
• I am trying to prover that my group is supersolvable. Because it has a faithful irreducible character the center is cyclic and then so is G''. This (I think) will help me trying to find a normal series in which every quotient is cyclic. – Joakim Færgeman Oct 25 '18 at 14:16

There are such groups with trivial centre. One such (possibly the smallest) is a group $$G$$ of order $$448$$ with the shape $$2^{3+3}:7$$. It has derived group $$G'$$ of order $$64$$, and $$G''$$ has order $$8$$ and is the unique minimal normal subgroup of $$G$$.
This is $$\tt{SmallGroup}(448,179)$$ in the databases in GAP and Magma. You can compute its character table, and it has two faithful irreducible characters of degree $$14$$.
Edit: In fact $$\tt{SmallGroup}(108,17)$$ with structure $$3^{1+2}:4$$ is a smaller example.