= Modular Heisenberg group
{title2=$H_p=UT_3(\mathbb F_p)$}
The modular Heisenberg group consists of upper-unitriangular $3\times3$ matrices over the <finite field> $\mathbb F_p$. In coordinates $M(x,y,z)$, multiplication is $M(x,y,z)M(x\prime,y\prime,z\prime)=M(x+x\prime,y+y\prime+xz\prime,z+z\prime)$, and its order is $p^3$. Its <center of a group> is $\{M(0,y,0)\}\cong C_p$; the <quotient group> by the centre is $C_p\times C_p$, represented by the $x,z$ coordinates. The elementary generators $a=M(1,0,0)$, $b=M(0,0,1)$, $c=M(0,1,0)$ satisfy $a^p=b^p=c^p=e$, $aba^{-1}b^{-1}=c$, and $c$ central. For $p=2$, $(ab)^2=c$, so the whole group does not have exponent two.
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