Groups of order 1728

This article gives information about, and links to more details on, groups of order 1728
See pages on algebraic structures of order 1728 | See pages on groups of a particular order

Statistics at a glance

The number 1728 has the following prime factorization:

$\!1728=2^{6}\cdot 3^{3}=64\cdot 27$

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.

Numbers of groups

Quantity	Value	Explanation
Total number of groups	47937
Number of abelian groups	33	equals the number of unordered integer partitions of 6 times the number of unordered integer partitions of 3 = 11 times 3. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.

GAP implementation

The order 1728 is part of GAP's SmallGroup library. Hence, any group of order 1728 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 1728 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:

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