Groups of order 560

This article gives information about, and links to more details on, groups of order 560
See pages on algebraic structures of order 560 | See pages on groups of a particular order

Statistics at a glance

The number 560 has prime factors 2, 5, and 7. The prime factorization is:

$\!560=2^{4}\cdot 5^{1}\cdot 7^{1}=16\cdot 5\cdot 7$

All groups of this order are solvable groups, and hence finite solvable groups.

GAP implementation

The order 560 is part of GAP's SmallGroup library. Hence, any group of order 560 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 560 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(560);

  There are 180 groups of order 560.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 70, 1 ].
     2 has Frattini factor [ 70, 2 ].
     3 has Frattini factor [ 70, 3 ].
     4 has Frattini factor [ 70, 4 ].
     5 has Frattini factor [ 140, 5 ].
     6 has Frattini factor [ 140, 6 ].
     7 - 31 have Frattini factor [ 140, 7 ].
     32 - 47 have Frattini factor [ 140, 8 ].
     48 - 63 have Frattini factor [ 140, 9 ].
     64 - 79 have Frattini factor [ 140, 10 ].
     80 - 87 have Frattini factor [ 140, 11 ].
     88 - 94 have Frattini factor [ 280, 32 ].
     95 - 101 have Frattini factor [ 280, 34 ].
     102 - 108 have Frattini factor [ 280, 35 ].
     109 - 135 have Frattini factor [ 280, 36 ].
     136 - 145 have Frattini factor [ 280, 37 ].
     146 - 155 have Frattini factor [ 280, 38 ].
     156 - 165 have Frattini factor [ 280, 39 ].
     166 - 169 have Frattini factor [ 280, 40 ].
     170 - 180 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.