Groups of order 30

This article gives information about, and links to more details on, groups of order 30
See pages on algebraic structures of order 30 | See pages on groups of a particular order

Statistics at a glance

The number 30 has prime factors 2, 3, and 5. The prime factorization is:

$30=2^{1}\cdot 3^{1}\cdot 5^{1}$

Square-free implies solvability-forcing, so all groups of order 30 are finite solvable groups. Moreover, every Sylow subgroup is cyclic implies metacyclic, so all groups of order 30 are in fact metacyclic groups.

The list

There are 4 groups of order 30:

Group	Second part of GAP ID	Abelian	Nilpotent	Direct Product
SmallGroup(30,1)	1	no	no	yes
SmallGroup(30,2)	2	no	no	yes
dihedral group:D30	3	no	no	no
cyclic group:Z30	4	yes	yes	yes

GAP implementation

The order 30 is part of GAP's SmallGroup library. Hence, any group of order 30 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 30 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(30);

  There are 4 groups of order 30.
    1 is of type S3x5.
    2 is of type D10x3.
    3 - 3 are of types 3:2+5:2.
    4 is of type c30.

  The groups whose order factorises in at most 3 primes
  have been classified by O. Hoelder. This classification is
  used in the SmallGroups library.

  This size belongs to layer 1 of the SmallGroups library.
  IdSmallGroup is available for this size.