Presentations for groups of prime-cube order

The discussion here applies in full to $$p \ge 3$$. The behavior for $$p = 2$$ is somewhat different. See presentations for groups of order 8 for more details.

Power-commutator presentations
Each of the power-commutator presentations uses three generators $$a_1,a_2,a_3$$. The power relations are of the form $$a_1^p = a_2^{\beta(1,2)} a_3^{\beta(1,3)}$$ where $$\beta(1,2), \beta(1,3)$$ are natural numbers that depend on the nature of the group, and $$a_2^p = a_3^{\beta(2,3)}$$ where $$\beta(2,3)$$ depends on the nature of the group. The commutator relation is of the form $$[a_1,a_2] = a_3^{\beta(1,2,3)}$$ where $$\beta(1,2,3)$$ depends on the nature of the group. Note that $$a_3^p$$, $$[a_1,a_3]$$, and $$[a_2,a_3]$$ are always the identity.

It turns out that for the isomorphism class of the final group, all the four values $$\beta(1,2),\beta(1,3),\beta(2,3),\beta(1,2,3)$$ matter only mod $$p$$. Thus, for simplicity, we assume that they are in the set $$\{ 0, 1, 2, \dots, p - 1 \}$$. Note that this is a general feature of power-commutator presentations.

Simplified power-commutator presentations
We here provide a single power-commutator presentation among the many possibilities.





Determining the isomorphism class from an arbitrary power-commutator presentation
Note that for the second and fourth groups, there are multiple sets of possible conditions, given in two separate rows within those groups. These are equivalent under permutations of the generators

The following algorithmic approach is a little faster:

We see now that each ID has a unique condition letter combination: