For odd $p$, a central product of two modular groups of order $p^3$ is a central product of a modular group with a Heisenberg group

Given: An odd prime $p$, two copies of $M_p$ with generators $a_i$ of order $p^2$ and $s_i$ of order $p$ satisfying $s_i{a}_i{s}_i^{-1}=a_i^{1+p}$, and the central product $E$ of the two along an isomorphism of their centres, with canonical images $x_i,y_i$ of $a_i,s_i$ and common central image $z$.

The modular group of order $p^3$ is $M_p=A{⋊}_{\alpha }B$ with $A=⟨a⟩$ of order $p^2$, $B=⟨s⟩$ of order $p$, and $sa{s}^{-1}=a^{1+p}$ (The modular group of order $p^3$ as a semidirect product $C_{p^2}⋊C_p$).

For groups $G,H$ with central subgroups $Z_1\le Z(G)$, $Z_2\le Z(H)$ and an isomorphism $\alpha :Z_1\to Z_2$, the central product $G{\circ }_{\alpha }H$ is the quotient of $G\times H$ by $N=\{(z,\alpha (z{)}^{-1}):z\in Z_1\}$ (The central product $G{\circ }_{\alpha }H$ of two groups along an isomorphism of central subgroups).

For $g,h\in G$ the commutator is $[g,h]:=gh{g}^{-1}{h}^{-1}$ (Commutators $[g,h]=gh{g}^{-1}{h}^{-1}$ and the commutator subgroup $[G,G]$).

$Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}$ (The center $Z(G)$ of a group).

For a finite group $G$, $\exp (G)=\min \{n\in \mathbb{N}:n>0\text{ and }{g}^n=e\text{ for every }g\in G\}$ (The exponent of a finite group).

$⟨S⟩$ is the smallest subgroup of $G$ containing $S$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).

$M_p$ is nonabelian of order $p^3$, extraspecial of exponent $p^2$, with $Z(M_p)=[M_p,M_p]=⟨a^p⟩$ (The modular group of order $p^3$ is extraspecial, of exponent $p^2$ when $p$ is odd).

A central product of two extraspecial $p$-groups identified along their centres is extraspecial of order $\mid E_1\mid \mid E_2\mid \mathrm{/}p$ (A central product of extraspecial $p$-groups identified along their centres is extraspecial).

The canonical maps into a central product are injective homomorphisms whose images commute elementwise, generate the product, and meet in the image of the identified subgroup (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

For an odd prime $p$ and a finite group $G$ with $[G,G]\le Z(G)$ of exponent dividing $p$, $(xy{)}^p=x^p{y}^p$ for all $x,y$ (For an odd prime $p$, the $p$-th power map is a homomorphism on a finite group whose derived subgroup is central of exponent dividing $p$).

An extraspecial $p$-group is nilpotent of class exactly two, its derived subgroup satisfies $P^{\mathrm{\prime }}=Z(P)$ and has order $p$, and every nonidentity commutator has order $p$ (An extraspecial $p$-group is nilpotent of class exactly two and its derived subgroup has order $p$).

If $[G,G]\le Z(G)$ then $[xy,w]=[x,w][y,w]$, $[x,yw]=[x,y][x,w]$, and $[x^n,y]=[x,y{]}^n=[x,y^n]$ for every integer $n$ (Commutator identities in a group whose derived subgroup is central).

If $[x,y]\mathrm{\ne }e$ in an extraspecial $p$-group $P$, then $⟨x,y⟩$ contains $Z(P)$, has order $p^3$, is nonabelian, and is extraspecial with centre $Z(P)$ (Two elements of an extraspecial $p$-group with nontrivial commutator generate an extraspecial subgroup of order $p^3$).

If $x,y\in P$ satisfy $[x,y]\mathrm{\ne }e$ and $F=⟨x,y⟩$, then $P=F{C}_P(F)$ and $F\cap C_P(F)=Z(P)$, and $C_P(F)$ is extraspecial of order $\mid P\mid \mathrm{/}{p}^2$ with centre $Z(P)$ when $\mid P\mid >p^3$ (Every extraspecial $p$-group is an internal central product of nonabelian subgroups of order $p^3$, The centralizer $C_G(H)$ of a subgroup).

For every prime $p$ there are exactly two nonabelian groups of order $p^3$ up to isomorphism; for odd $p$ they are $H_p$, of exponent $p$, and $M_p$, of exponent $p^2$ (For each prime there are exactly two nonabelian groups of order $p^3$ up to isomorphism).

Subgroups form an internal central product of $G$ if and only if the multiplication map from their direct product is a surjective homomorphism; for two factors $G\cong G_1{\circ }_{\mathrm{id}}{G}_2$ along the identity of $G_1\cap G_2$ (Internal central products are the images of external ones).

For all $m,n\in \mathbb{Z}$ one has $(g^m{)}^n=g^{mn}$ (Exponent laws in a group: $g^{m+n}=g^m{g}^n$ and $(g^m{)}^n=g^{mn}$ for all $m,n\in \mathbb{Z}$, and $(gh{)}^n=g^n{h}^n$ when $g$ and $h$ commute).