Suppose $G=\langle a,b \mid [a,b]^{p^\gamma}=[a,b,a]=[a,b,b]=1, a^{p^{\alpha}}=[a,b]^{p^\rho}, b^{p^{\beta}}=[a,b]^{p^\sigma}\rangle$, where $\alpha>\beta\geq \gamma\geq1$ and $0\leq\sigma<\rho<\min\{\gamma,\sigma+\alpha-\beta\}$. From these conditions, we obtain $\alpha>\beta\geq \gamma>\rho>\sigma\geq0$.
If we consider the following group, not in the category of the above groups (since $\alpha=\beta\;\text{and}\;\sigma=\gamma$ here), $G_1=\langle a,b \mid [a,b]^{27}=[a,b,a]=[a,b,b]=1, a^{81}=[a,b]^{9}, b^{81}=[a,b]^{27}=1\rangle$ , we have $G_1 \cong (\langle c\rangle \times \langle a\rangle)\rtimes \langle b\rangle, \;\text{with}\; [a, b] = a^9c, [b, c] = [a,b]^9,|a| =243, |b| = 81, |c| =9, |[a, b]| = 27; \text{i.e.}, G_1\cong(\mathbb{Z_{243}}\times \mathbb{Z_{9}})\rtimes\mathbb{Z_{81}},$ using 1, page 21,(iii),with $(t+w-v,u,v;w,v)=(4,4,3,2,3)$.
My question is: Can we write a group $G$ as a semidirect product of some groups, such as $G_1$ defined in 1 ? If yes, what will be those generators and relations?
I really appreciate any help you can provide.