Groups of order 84

This article gives information about, and links to more details on, groups of order 84
See pages on algebraic structures of order 84 | See pages on groups of a particular order

Statistics at a glance

The number 84 has the prime factorization:

$\!84=2^{2}\cdot 3^{1}\cdot 7^{1}=4\cdot 3\cdot 7$

GAP implementation

The order 84 is part of GAP's SmallGroup library. Hence, any group of order 84 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 84 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(84);

  There are 15 groups of order 84.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 42, 1 ].
     2 has Frattini factor [ 42, 2 ].
     3 has Frattini factor [ 42, 3 ].
     4 has Frattini factor [ 42, 4 ].
     5 has Frattini factor [ 42, 5 ].
     6 has Frattini factor [ 42, 6 ].
     7 - 15 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.