# Classification of groups of prime-fourth order

## Statement

Let $p$ be a prime number. Then, the groups of order $p^4$ can be classified as follows, with slightly different classifications for the cases $p = 2$, $p = 3$, and $p \ge 5$. The case $p = 2$ is different even in so far as the number of possible groups is concerned. The cases $p = 3$ and $p \ge 5$ have minor differences with each other.

Group property Number of isomorphism classes of groups case $p = 2$, so order $16$ Number of isomorphism classes of groups case odd prime
abelian group 5 5
nilpotency class exactly two, i.e., class two group that is non-abelian 6 6
nilpotency class exactly three 3 4
Total 14 15

### The five abelian groups

The nature and classification of the five abelian groups of order $p^4$ is the same for both the $p = 2$ and odd $p$ cases; the abelian groups are classified by the set of unordered integer partitions of the number 4.

Partition of 4 Corresponding abelian group (in general) Corresponding abelian group case $p = 2$ GAP ID (2nd part) case $p = 2$ GAP ID (2nd part) case odd $p$
4 cyclic group of prime-fourth order cyclic group:Z16 1 1
3 + 1 direct product of cyclic group of prime-cube order and cyclic group of prime order direct product of Z8 and Z2 5 5
2 + 2 direct product of cyclic group of prime-square order and cyclic group of prime-square order direct product of Z4 and Z4 2 2
2 + 1 + 1 direct product of cyclic group of prime-square order and elementary abelian group of prime-square order direct product of Z4 and V4 10 11
1 + 1 + 1 + 1 elementary abelian group of prime-fourth order elementary abelian group:E16 14 15