The agemo series of a $p$-group $G$, where $p$ is a prime number, is the descending normal series

The AgemoSeries( G ) command constructs the agemo series of a group G, which must be a finite $p$-group, for some prime $p$.

The omega series of a finite $p$-group $G$ is the ascending normal series

The OmegaSeries( G ) command constructs the omega series of a finite $p$-group G.

The group G must be an instance of a permutation group.

Both the agemo and omega series of G are represented by a NormalSeries object which admits certain operations common to all normal series.  See GroupTheory[Series].

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{DihedralGroup}\left(8\right)$

$G≔{\mathbf{D}}_8$

$\mathrm{as}≔\mathrm{AgemoSeries}\left(G\right)$

$\mathrm{as}≔{\mathbf{D}}_8◃{\&Agemo;}_1\left({\mathbf{D}}_8\right)◃{\&Agemo;}_2\left({\mathbf{D}}_8\right)◃{\&Agemo;}_3\left({\mathbf{D}}_8\right)$

$\mathrm{numelems}\left(\mathrm{as}\right)$

$4$

$\mathrm{os}≔\mathrm{OmegaSeries}\left(G\right)$

$\mathrm{os}≔⟨⟩◃{\mathbf{D}}_8$

$\mathrm{numelems}\left(\mathrm{os}\right)$

$2$