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

Question

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?

Answer 1

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.