Sylow subgroup of holomorph of cyclic group of odd prime-cube order

Definition
Let $$p$$ be an odd prime number. This group is defined as the $$p$$-Sylow subgroup of the holomorph of the cyclic group of order $$p^3$$. Equivalently, it is the semidirect product of the cyclic group of order $$p^3$$ and the cyclic group of order $$p^2$$, where the generator of the latter acts on the former by the $$(p+1)$$-power map.

It can be given by the explicit presentation:

$$G := \langle a,b \mid a^{p^3} = b^{p^2} = e, bab^{-1} = a^{p + 1}\rangle$$

Note that the case $$p = 2$$ is different, because the automorphism structure of the cyclic group of order $$2^3$$ is different, with a non-cyclic 2-Sylow subgroup of automorphisms. See holomorph of Z8 for details on that group.

Facts

 * For $$p$$ odd, this is a finite p-group that is not characteristic in any finite p-group properly containing it.

GAP implementation
Here, p is the previously assigned value of the prime number, that we assume here to be odd. You can replace p by an actual numerical value of a prime or precede these commands by an assignment p :=  the value.

gap> C := CyclicGroup(p^3);  gap> A := AutomorphismGroup(C);  gap> S := SylowSubgroup(A,p); gap> G := SemidirectProduct(S,C); 