Heisenberg group over a prime field ($U{T}_3({\mathbb{F}}_p)$)

Upper-unitriangular three-by-three matrices over the prime finite field form the finite analogue of the real Heisenberg group. In coordinates their multiplication is $(a,b,c)(a^{\prime },b^{\prime },c^{\prime })=(a+a^{\prime },b+b^{\prime },c+c^{\prime }+a{b}^{\prime })$. The order is $p^3$, and $(a,b,c{)}^k=(ka,kb,kc+\left(\genfrac{}{}{0ex}{}{k}{2}\right)ab)$. For odd $p$, every nonidentity element has order $p$, but the group is nonabelian. For $p=2$ the exponent statement changes, so odd characteristic is essential for the nonisomorphic Gassmann equivalent regular subgroups construction.