# Groups of order 1920

## Contents

See pages on algebraic structures of order 1920| See pages on groups of a particular order

## Statistics at a glance

The number 1920 has prime factors 2,3, and 5. The prime factorization is: $\! 1920 = 2^7 \cdot 3 \cdot 5 = 128 \cdot 3 \cdot 5$

Quantity Value Explanation
Total number of groups up to isomorphism 241004
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 15 (Number of abelian groups of order $2^7$) times (Number of abelian groups of order $3^1$) times (Number of abelian groups of order $5^1$) = (number of unordered integer partitions of 7) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) = $15 \times 1 \times 1 = 15$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 2328 (Number of groups of order 128) times (Number of groups of order 3) times (Number of groups of order 5) = $2328 \times 1 \times 1 = 2328$. See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.
Number of supersolvable groups (i.e., finite supersolvable groups) up to isomorphism PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Number of solvable groups (i.e., finite solvable groups) up to isomorphism PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] See note on non-solvable groups
Number of non-solvable groups up to isomorphism PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] All the non-solvable groups have alternating group:A5 as the unique simple non-abelian composition factor and five cyclic group:Z2s as the other composition factors
Number of simple groups up to isomorphism 0
Number of quasisimple groups up to isomorphism 0
Number of almost simple groups up to isomorphism 0
Number of perfect groups up to isomorphism 7

## GAP implementation

The order 1920 is part of GAP's SmallGroup library. Hence, any group of order 1920 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 1920 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(1920);

There are 241004 groups of order 1920.
There are sorted using Hall subgroups.
1 - 2328 are the nilpotent groups.
2329 - 236344 have a normal Hall (3,5)-subgroup.
236345 - 240416 are solvable without normal Hall (3,5)-subgroup.
240417 - 241004 are not solvable.

This size belongs to layer 6 of the SmallGroups library.
IdSmallGroup is available for this size.