Groups of order 448

This article gives information about, and links to more details on, groups of order 448
See pages on algebraic structures of order 448 | See pages on groups of a particular order

Statistics at a glance

The number 448 has prime factors 2 and 7. The prime factorization is:

$\!448=2^{6}\cdot 7=64\cdot 7$

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.

Quantity	Value	Explanation
Number of groups up to isomorphism	1396
Number of abelian groups up to isomorphism	11	Equals the number of unordered integer partitions of $6$ times the number of unordered integer partitions of $1$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.

See also

• Classification of groups of order sixty-four times a prime

GAP implementation

The order 448 is part of GAP's SmallGroup library. Hence, any group of order 448 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 448 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(448);

  There are 1396 groups of order 448.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 14, 1 ].
     2 has Frattini factor [ 14, 2 ].
     3 - 124 have Frattini factor [ 28, 3 ].
     125 - 177 have Frattini factor [ 28, 4 ].
     178 - 179 have Frattini factor [ 56, 11 ].
     180 - 781 have Frattini factor [ 56, 12 ].
     782 - 918 have Frattini factor [ 56, 13 ].
     919 has Frattini factor [ 112, 41 ].
     920 - 1293 have Frattini factor [ 112, 42 ].
     1294 - 1361 have Frattini factor [ 112, 43 ].
     1362 - 1364 have Frattini factor [ 224, 195 ].
     1365 - 1384 have Frattini factor [ 224, 196 ].
     1385 - 1391 have Frattini factor [ 224, 197 ].
     1392 - 1396 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.