Classification of groups of prime-fourth order

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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