# Groups of prime-fourth order

This article is about the groups of prime-fourth order for an odd prime number, i.e., the groups of order $p^4$ where $p$ is an odd prime. The special case $p = 2$ is somewhat different -- see groups of order 16 for a summary of information on these groups.

Among the odd primes $p$, the case $p = 3$ is slightly different from the other primes.

## Statistics at a glance

### Group counts

Quantity Value case $p = 2$ Value case $p = 3$ Value case $p \ge 5$ Explanation
Total number of groups 14 15 15 See classification of groups of order 16, classification of groups of prime-fourth order for odd prime
Number of abelian groups 5 5 5 See classification of finite abelian groups and structure theorem for finitely generated abelian groups.. In this case, the number of unordered integer partitions of 4 equals 5.
Number of groups of nilpotency class exactly two 6 6 6
Number of groups of nilpotency class exactly three 3 4 4

## Particular cases

Prime $p$ $p^4$ Number of groups Information on groups of order $p^4$
2 16 14 groups of order 16 -- behaves quite differently from the others.
3 81 15 groups of order 81 -- behaves somewhat differently from the others.
5 625 15 groups of order 625
7 2401 15 groups of order 2401
11 14641 15 groups of order 14641