Groups of order 210

This article gives information about, and links to more details on, groups of order 210
See pages on algebraic structures of order 210 | See pages on groups of a particular order

Statistics at a glance

The number 210 has prime factors 2, 3, 5, and 7. The prime factorization is:

$\ 210=2^{1}\cdot 3^{1}\cdot 5^{1}\cdot 7^{1}=2\cdot 3\cdot 5\cdot 7$

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

See also

• Classification of groups of order a product of four distinct primes

GAP implementation

The order 210 is part of GAP's SmallGroup library. Hence, any group of order 210 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 210 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(210);

  There are 12 groups of order 210.
  They are sorted by their Frattini factors.
     1 - 12 have trivial Frattini subgroup.

  For the selection functions the values of the following attributes
  are precomputed and stored:
     IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
     LGLength, FrattinifactorSize and FrattinifactorId.

  This size belongs to layer 2 of the SmallGroups library.
  IdSmallGroup is available for this size.