# Congruence condition summary for groups of prime-cube order

This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of prime-cube order.

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## Case of the prime 2

We first provide the congruence condition summary for groups of order 8, i.e., the case where .

There are five groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8. There are thus possible collections of groups. Instead of listing all 31, we simply note the ones that *do* satisfy a universal congruence condition or a congruence condition to an interesting restricted class:

## Case of the prime 3

We now provide the congruence condition summary for groups of order 27, i.e., the case .

## Case of primes greater than 3

For any prime , the congruence condition summary for groups of order looks the same. It is given below.

First, note that there are five groups of order . The three abelian groups are cyclic group of prime-cube order, direct product of cyclic group of prime-square order and cyclic group of prime order, and elementary abelian group of prime-cube order. The two non-abelian groups are prime-cube order group:U(3,p) (this has exponent ) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (this has exponent ).