# Groups of order 48

## Contents

See pages on algebraic structures of order 48| See pages on groups of a particular order

Information type Page summarizing information for groups of order 48
element structure (element orders, conjugacy classes, etc.) element structure of groups of order 48
subgroup structure subgroup structure of groups of order 48
linear representation theory linear representation theory of groups of order 48
projective representation theory of groups of order 48
modular representation theory of groups of order 48
endomorphism structure, automorphism structure endomorphism structure of groups of order 48
group cohomology group cohomology of groups of order 48

## Statistics at a glance

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

### Factorization and useful forms

The number 48 has prime factors 2 and 3, and factorization: $48 = 2^4 \cdot 3^1 = 16 \cdot 3$

Other expressions for this number are: $48 = (3^2 - 1)(3^2 - 3) = 2(4!) = \frac{4}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} - 1}$

### Group counts

Quantity Value Explanation
Total number of groups 52
Number of abelian groups 5 (number of abelian groups of order $2^4$) times (number of abelian groups of order $3^1$) = (number of unordered integer partitions of 4) times (number of unordered integer partitions of 1) = $5 \times 1 = 5$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups 14 (number of groups of order 16) times (number of groups of order 3) = $14 \times 1 = 14$. 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 52 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.
Number of simple groups 0 Follows from all groups of this order being solvable.

## Sylow subgroups

### 2-Sylow subgroups

Here is the occurrence summary:

Group of order 16 GAP ID (second part) Number of groups of order 48 in which it is a 2-Sylow subgroup List of these groups Second part of GAP ID of these groups
cyclic group:Z16 1 2 1, 2
direct product of Z4 and Z4 2 3 3, 11, 20
SmallGroup(16,3) 3 4 14, 19, 21, 30
nontrivial semidirect product of Z4 and Z4 4 3 12, 13, 22
direct product of Z8 and Z2 5 3 4, 9, 23
M16 6 3 5, 10, 24
dihedral group:D16 7 3 7, 15, 25
semidihedral group:SD16 8 5 6, 16, 17, 26, 29
generalized quaternion group:Q16 9 4 8, 18, 27, 28
direct product of Z4 and V4 10 4 31, 35, 42, 44
direct product of D8 and Z2 11 5 36, 38, 43, 45, 48
direct product of Q8 and Z2 12 4 32, 34, 40, 46
central product of D8 and Z4 13 5 33, 37, 39, 41, 47
elementary abelian group:E16 14 4 49, 50, 51, 52

## GAP implementation

The order 48 is part of GAP's SmallGroup library. Hence, any group of order 48 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 48 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(48);

There are 52 groups of order 48.
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 - 19 have Frattini factor [ 12, 4 ].
20 - 27 have Frattini factor [ 12, 5 ].
28 - 30 have Frattini factor [ 24, 12 ].
31 - 33 have Frattini factor [ 24, 13 ].
34 - 43 have Frattini factor [ 24, 14 ].
44 - 47 have Frattini factor [ 24, 15 ].
48 - 52 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.