# Groups of order 1792

## Contents

This article gives information about, and links to more details on, groups of order 1792
See pages on algebraic structures of order 1792| See pages on groups of a particular order

## Statistics at a glance

The number 1792 has prime factors 2 and 7. The prime factorization is: $\! 1792 = 2^8 \cdot 7 = 256 \cdot 7$

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-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.

## GAP implementation

The order 1792 is part of GAP's SmallGroup library. Hence, any group of order 1792 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 1792 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(1792);

There are 1083553 groups of order 1792.
They are sorted by normal Sylow subgroups.
1 - 56092 are the nilpotent groups.
56093 - 1083472 have a normal Sylow 7-subgroup
with centralizer of index 2.
1083473 - 1083549 have a normal Sylow 2-subgroup.
1083550 - 1083553 have no normal Sylow subgroup.

This size belongs to layer 3 of the SmallGroups library.
IdSmallGroup is available for this size.