# Groups of order 96

## Contents

See pages on algebraic structures of order 96| See pages on groups of a particular order

Information type Page summarizing information for groups of order 96
element structure (element orders, conjugacy classes, etc.) element structure of groups of order 96
subgroup structure subgroup structure of groups of order 96
linear representation theory linear representation theory of groups of order 96
projective representation theory of groups of order 96
modular representation theory of groups of order 96
endomorphism structure, automorphism structure endomorphism structure of groups of order 96
group cohomology group cohomology of groups of order 96

## Statistics at a glance

To understand these in a broader context, see groups of order 3.2^n

### Factorization and useful forms

The number 96 has prime factors 2 and 3, and factorization: $96 = 2^5 \cdot 3^1 = 32 \cdot 3$

Other expressions for this number are: $96 = 2(3^2 - 1)(3^2 - 3)$

### Group counts

Quantity Value List/comment
Total number of groups up to isomorphism 231
Number of abelian groups 7 ((number of abelian groups of order $32 = 2^5$)) times ((number of abelian groups of order $3^1$) = (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.
Number of nilpotent groups 51 (number of groups of order 32) times (number of groups of order 3) = $51 \times 1 = 51$. 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 231 Since there are only two prime factors, and order has only two prime factors implies solvable, all groups of this order are solvable groups (specifically, finite solvable groups).
Number of simple groups 0 Follows from all groups of the order being solvable.

## GAP implementation

The order 96 is part of GAP's SmallGroup library. Hence, any group of order 96 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 96 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(96);

There are 231 groups of order 96.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 - 44 have Frattini factor [ 12, 4 ].
45 - 63 have Frattini factor [ 12, 5 ].
64 - 67 have Frattini factor [ 24, 12 ].
68 - 74 have Frattini factor [ 24, 13 ].
75 - 160 have Frattini factor [ 24, 14 ].
161 - 184 have Frattini factor [ 24, 15 ].
185 - 195 have Frattini factor [ 48, 48 ].
196 - 202 have Frattini factor [ 48, 49 ].
203 - 204 have Frattini factor [ 48, 50 ].
205 - 219 have Frattini factor [ 48, 51 ].
220 - 225 have Frattini factor [ 48, 52 ].
226 - 231 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.