Presentations for groups of prime-cube order

This article gives specific information, namely, presentations, about a family of groups, namely: groups of prime-cube order.
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The discussion here applies in full to $p\geq 3$. 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.

Group	Second part of GAP ID (GAP ID is (p^3,2nd part)	Nilpotency class	Minimum size of generating set	Prime-base logarithm of exponent	$\beta (1,2)$	$\beta (1,3)$	$\beta (2,3)$	$\beta (1,2,3)$	full power-commutator presentation
cyclic group of prime-cube order	1	1	1	3	1	0	1	0	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=a_{2},a_{2}^{p}=a_{3},a_{3}^{p}=e,[a_{1},a_{2}]=e,[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
direct product of cyclic group of prime-square order and cyclic group of prime order	2	1	2	2	0	1	0	0	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=a_{3},a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=e,[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
prime-cube order group:U(3,p)	3	2	2	1	0	0	0	1	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=e,a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=a_{3},[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	2	2	2	0	1	0	1	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=a_{3},a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=a_{3},[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
elementary abelian group of prime-cube order	5	1	3	1	0	0	0	0	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=e,a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=e,[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$

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

Group	Second part of GAP ID (GAP ID is (p^3,2nd part)	Nilpotency class	Minimum size of generating set	Condition on $\beta (1,2)$	Condition on $\beta (1,3)$	Condition on $\beta (2,3)$	Condition on $\beta (1,2,3)$
cyclic group of prime-cube order	1	1	1	nonzero	arbitrary	nonzero	zero
direct product of cyclic group of prime-square order and cyclic group of prime order	2	1	2	arbitrary nonzero zero nonzero	nonzero arbitrary zero zero	zero zero nonzero nonzero	zero zero zero zero
prime-cube order group:U(3,p)	3	2	3	zero	zero	zero	nonzero
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	2	2	zero zero	zero nonzero	nonzero zero	nonzero nonzero
elementary abelian group of prime-cube order	5	1	3	zero	zero	zero	zero

The following algorithmic approach is a little faster:

Condition letter	Condition detail	If yes, then possible GAP IDs	If no, then possible GAP IDs
A	$\beta (1,2,3)=0$	1,2,5 (abelian groups)	3,4 (non-abelian groups)
B	All: $\beta (1,2)=0,\beta (2,3)=0,\beta (1,3)=0$	3,5 (exponent $p$ groups)	1,2,4
C	All: $\beta (1,2)\neq 0$, $\beta (2,3)\neq 0$	1,4	2,3,5

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

Group	Second part of GAP ID (GAP ID is (p^3,2nd part)	Nilpotency class	Minimum size of generating set	Condition letter combination
cyclic group of prime-cube order	1	1	1	satisfies A and C but not B
direct product of cyclic group of prime-square order and cyclic group of prime order	2	1	2	satisfies A but not B or C
prime-cube order group:U(3,p)	3	2	3	satisfies B but not A or C
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	2	2	satisfies C but not A or B
elementary abelian group of prime-cube order	5	1	3	satisfies A and B but not C (in fact, it suffices to check A and B and ignore C)