Groups of order 56

This article gives information about, and links to more details on, groups of order 56
See pages on algebraic structures of order 56 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 56. The prime factorization of 56 is $56=2^{3}\cdot 7$. There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Statistics at a glance

Quantity	Value
Total number of groups	13
Number of abelian groups	3
Number of nilpotent groups	5
Number of solvable groups	13
Number of simple groups	0

Classification

Further information: Classification of groups of order 56

The list

There are 13 groups of order 56:

Group	Second part of GAP ID	Abelian	Nilpotent	Direct Product
Nontrivial semidirect product of Z7 and Z8	1	no	no	no
Cyclic group:Z56	2	yes	yes	no
Dicyclic group:Dic56	3	no	no	no
Direct product of Z4 and D14	4	no	no	yes
Dihedral group:D56	5	no	no	no
Direct product of Z2 and Dic28	6	no	no	yes
Nontrivial semidirect product of Z7 and D8	7	no	no	no
Direct product of Z2 and Z28	8	yes	yes	yes
Direct product of Z7 and D8	9	no	yes	yes
Direct product of Z7 and Q8	10	no	yes	yes
General affine group:GA(1,8)	11	no	no	no
Direct product of V4 and D14	12	no	no	yes
Direct product of V4 and Z14	13	yes	yes	yes

Arithmetic functions

Group	Second part of GAP ID	Nilpotency class
Nontrivial semidirect product of Z7 and Z8	1	-
Cyclic group:Z56	2	1
Dicyclic group:Dic56	3	-
Direct product of Z4 and D14	4	-
Dihedral group:D56	5	-
Direct product of Z2 and Dic28	6	-
Nontrivial semidirect product of Z7 and D8	7	-
Direct product of Z2 and Z28	8	1
Direct product of Z7 and D8	9	2
Direct product of Z7 and Q8	10	2
General affine group:GA(1,8)	11	-
Direct product of V4 and D14	12	-
Direct product of V4 and Z14	13	1

Minimal order attaining number

$56$ is the smallest number such that there are precisely $13$ groups of that order up to isomorphism. That is, the value of the minimal order attaining function at $13$ is $56$.

GAP implementation

The order 56 is part of GAP's SmallGroup library. Hence, any group of order 56 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 56 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(56);

  There are 13 groups of order 56.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 14, 1 ].
     2 has Frattini factor [ 14, 2 ].
     3 - 7 have Frattini factor [ 28, 3 ].
     8 - 10 have Frattini factor [ 28, 4 ].
     11 - 13 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.