The modular group of order $p^3$ as a semidirect product $C_{p^2}⋊C_p$

Let $p$ be a prime, let $A=⟨a⟩$ be a cyclic group of order $p^2$ and let $B=⟨s⟩$ be a cyclic group of order $p$ (Every cyclic group is isomorphic to $(\mathbb{Z},+)$ or to $(\mathbb{Z}\mathrm{/}n,+)$ for its finite order $n\ge 1$). Define

$\alpha :B⟶\mathrm{Aut}(A),{\alpha }_{s^i}(x)=x^{(1+p{)}^i}\mathrm{.}$

Why this is well defined and is an action by automorphisms. Each $x\mapsto x^{1+p}$ is an automorphism of $A$ of order $p$ (Raising to the power $1+p$ is an automorphism of order $p$ of a cyclic group of order $p^2$), so ${\alpha }_{s^i}$ is the $i$-th power of that automorphism and is an automorphism. If $s^i=s^j$ then $p$ divides $i-j$, and the $p$-th power of that automorphism is the identity, so ${\alpha }_{s^i}={\alpha }_{s^j}$. Finally ${\alpha }_{s^i}\circ {\alpha }_{s^j}={\alpha }_{s^{i+j}}$, so $\alpha$ is a homomorphism into $\mathrm{Aut}(A)$ and hence an action by automorphisms (An action of a group $H$ on a group $N$ by automorphisms, Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$).

The modular group of order $p^3$ is the external semidirect product ( The external semidirect product $N{⋊}_{\alpha }H$, The semidirect-product multiplication makes $N\times H$ a group)

$M_p:=A{⋊}_{\alpha }B\mathrm{.}$

Writing $a$ and $s$ for the canonical images of the two generators (The canonical copy of $N$ is normal, the canonical copy of $H$ is a complement, and conjugation induces the action), the group is generated by $a$ and $s$ subject to

$a^{p^2}=1,s^p=1,sa{s}^{-1}=a^{1+p}\mathrm{.}$

Craven writes ${\mathrm{Mod}}_n(p)$ for the analogous group of order $p^n$ and reserves $p_{+}^{1+2}$ for the exponent-$p$ group; only the case $n=3$ is built here. Its relation uses the standard order-$p$ power automorphism $a\mapsto a^{1+p}$, the first nonidentity power automorphism congruent to the identity modulo $p$.

At $p=2$ the relation reads $sa{s}^{-1}=a^3=a^{-1}$, so the action is inversion and $M_2$ is the generalized dihedral group of a cyclic group of order four ( $\mathrm{Dih}(C_n)=C_n⋊C_2$ with inversion action has order $2n$ and the dihedral relations).