Groups of prime-fourth order

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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