Presentation of group $(\mathbb{Z}_p\times \mathbb{Z}_p )\rtimes \mathbb{Z}_p$ for odd prime $p.$

Question

I like to find presentation of the finite non Abelian group $$(\mathbb{Z}_p\times \mathbb{Z}_p )\rtimes \mathbb{Z}_p$$ of order $p^3$, where $p$ is an odd prime. According to me its presentation is $$\langle a,b,c: a^p=b^p=c^p=1, ab=ba, ac=ca, cb=abc\rangle$$ Please suggest me. Thanks.

Answer 1

Groupprops: Groups of prime-cube order shows the same result as yours:

1. There are 5 groups (up to isomorphism) of order $p^3$ for odd prime $p$.

2. Two of them are non-abelian, and one is the semidirect product of $\mathbb{Z}_p^2$ and $\mathbb{Z}_p$, that is not you wanted.

3. The other non-abelian group is the prime-cube order group: $U(3,p)$ and its presentation is $$ \langle a_1,a_2,a_3 \mid a_1^p=e, a_2^p=e, a_3^p=e, [a_1,a_2]=a_3, [a_1,a_3]=e, [a_2,a_3]=e \rangle $$ that is exactly what you wanted.