[erfanian@wali.um.ac.ir: [GAP Forum] looking for a group]

[Mike Newman]

I have been asked to post my private response to the Forum.

This has made no use at all of GAP.
It would be interesting to know how someone who can't drag examples from
thier personal archive might use GAP to find some.

Mike Newman

Message from erfanian <erfanian at wali.um.ac.ir> -----

Dear All,
I am looking for an example of a group $G$ with the property that $G/Z(G)$
is a p-elementary abelian of rank $k\geq 3$ and for every elements $x \in
G\Z(G)$ we have $[G : C_G(x)}=p$. I will be more grateful for any
comments.

Response:

The extra-special groups are examples.
These are groups with class 2 whose centre and commutator subgroup
coincide and have order p.
The finite extra-special groups are central products of the non-abelian
groups with order p^3.
The central product of an extra-special group and an abelian group is
also an example.
These are the only finite examples.