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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?

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    $\begingroup$ Any finite group with a unique minimal normal subgroup must have a faithful irreducible character, so your final condition is redundant. $\endgroup$ – Derek Holt Oct 25 '18 at 8:27
  • $\begingroup$ 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)? $\endgroup$ – Geoff Robinson Oct 25 '18 at 9:17
  • $\begingroup$ 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. $\endgroup$ – Joakim Færgeman Oct 25 '18 at 14:16
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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.

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