[GAP Forum] a question about p-groups

[Siddhartha Sarkar]

Dear Gap Forum,

Is it possible to generate a finite p-group G such that 

(1) $G/G_2$ is isomorphic to Z/pZ X Z/pZ
(2) $G_2/G_3$ is isomorphic to Z/pZ
(3) $G_2$ is of exponent p
(4) If g belong to G not $G_2$, then order(g) is bigger than p.

Here $G_2$ and $G_3$ are respectively the second and third term
in the lower central series.

The most natural to think is p-groups of maximal class. But upto
order p^p, I think there is no such group. 

with regards,
Siddhartha Sarkar

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Siddhartha Sarkar
Research Scholar
Dept. of Mathematics
Harish Chandra Research Institute
Chhatnag Road,Jhusi
Allahabad-211019.
India.
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