Center of SmallGroup(16,3)

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) Klein four-group and the group is (up to isomorphism) SmallGroup(16,3) (see subgroup structure of SmallGroup(16,3)).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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This article concerns the center of the group SmallGroup(16,3). Call the group SmallGroup(16,3) $G$ on this page.

SmallGroup(16,3) can be defined using the following presentation:

$G:=\langle a,b,c\mid a^{4}=b^{2}=c^{2}=e,ab=ba,bc=cb,cac^{-1}=ab\rangle$. It is the unique non-abelian semidirect product $(Z_{2}\times Z_{2})\rtimes Z_{4}$ (i.e. of the Klein four-group and cyclic group:Z4 up to isomorphism, and a semidirect product $(Z_{4}\times Z_{2})\rtimes Z_{2}$.

The subgroup $\langle a^{2},b\rangle$ is the center, as well as the first agemo subgroup, and Frattini subgroup. It is isomorphic to the Klein four-group.

The center is a normal subgroup. The quotient group $G/Z(G)$ is isomorphic to Klein four-group.