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Let $p$ be a prime. A $p$-group $G$ is defined to be semi-extraspecial if forevery maximal subgroup $N$ in $Z(G)$ the quotient $G/N$ is a an extraspecialgroup. In addition, we say that $G$ is ultraspecial if $G$ is semi-extraspecialand $|G:G'| = |G'|^2$. In this paper, we prove that every $p$-group ofnilpotence class $2$ is isomorphic to a subgroup of some ultraspecial group.Given a prime $p$ and a positive integer $n$, we provide a framework toconstruct of all the ultraspecial groups order $p^{3n}$ that contain an abeliansubgroup of order $p^{2n}$. In the literature, it has been proved that everyultraspecial group $G$ order $p^{3n}$ with at least two abelian subgroups oforder $p^{2n}$ can be associated to a semifield. We provide a generalization ofsemifield, and then we show that every semi-extraspecial group $G$ that is theproduct of two abelian subgroups can be associated with this generalization ofsemifield.

Semi-extraspecial groups with an abelian subgroup of maximal possible order