I'm working on a group for my master's thesis and I've found a fairly precise characterisation of it, but I can't see how to describe it as a combination of standard groups. I have characterised it as $$(\alpha^{i_1}\gamma^{i_2}L_0, \alpha^{i_2}\gamma^{i_3}L_0, \alpha^{i_3}\gamma^{i_1}L_0),$$
where both $\alpha$ and $\gamma$ are of degree 3 and $L_0$ is a group isomorphic to $C_3 \times C_3$.
(In other words an element of this group is characterised by the three numbers $i_1$, $i_2$ and $i_3$, and by three elements in $L_0$, giving the group $3^3*9^3 = 3^9$ elements.
Has anyone got an idea how to better describe this?
EDIT: As mentioned below I've given much too little information about this group. The group is a subgroup of $C_3 \wr C_3$, where $\alpha$ and $\gamma$ are the generators of the respective cyclic groups.
$L_0$ is a normal subgroup of $C_3 \wr C_3$, and more precisely of the subgroup of elements whose $\alpha$-coordinate is nil ($C_3 \wr C_3 \cong C_3 \ltimes C_3^3$, and what I'm saying is that $L_0$ is inside the $\{1\} \ltimes C_3^3$ part).
The group I'm investigating in general is the group generated by $(\alpha, 1, \gamma)$, $(\gamma, \alpha, 1)$ and $(1, \gamma, \alpha)$, and I have managed to reduce it to the above characterisation.