# Groups of order 160

## Contents

See pages on algebraic structures of order 160| See pages on groups of a particular order

## Statistics at a glance

The number 160 has prime factorization $160 = 2^5 \cdot 5$. There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 160 are solvable groups (specifically, finite solvable groups).

Quantity Value Explanation
number of groups up to isomorphism 238
number of abelian groups up to isomorphism 7 (number of abelian groups of order $2^5$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of 5) times (number of unordered integer partitions of 1) = $7 \times 1 = 7$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
number of nilpotent groups up to isomorphism 51 (number of groups of order 32) times (number of groups of order 5) = $51 \times 1 = 51$. 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 solvable groups up to isomorphism 238 There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 160 are solvable groups (specifically, finite solvable groups).
number of simple groups up to isomorphism 0 All groups of this order are solvable, so there cannot be any simple groups.

## GAP implementation

The order 160 is part of GAP's SmallGroup library. Hence, all groups of order 160 can be constructed using the SmallGroup function and have group IDs. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Here is GAP's summary information about how it stores groups of this order:

gap> SmallGroupsInformation(160);

There are 238 groups of order 160.
They are sorted by their Frattini factors.
1 has Frattini factor [ 10, 1 ].
2 has Frattini factor [ 10, 2 ].
3 has Frattini factor [ 20, 3 ].
4 - 44 have Frattini factor [ 20, 4 ].
45 - 63 have Frattini factor [ 20, 5 ].
64 - 88 have Frattini factor [ 40, 12 ].
89 - 174 have Frattini factor [ 40, 13 ].
175 - 198 have Frattini factor [ 40, 14 ].
199 has Frattini factor [ 80, 49 ].
200 - 212 have Frattini factor [ 80, 50 ].
213 - 227 have Frattini factor [ 80, 51 ].
228 - 233 have Frattini factor [ 80, 52 ].
234 - 238 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.