Bureaucrats, emailconfirmed, Administrators

38,910

edits
Jump to: navigation, search

no edit summary

==Statistics at a glance==

The number 48 has prime factorization <math>48 = 2^4 \cdot 3</math>. {{only two prime factors hence solvable}}

{| class="sortable" border="1"

! Quantity !! Value!! Explanation

|-

| Total number of groups || [[count::52]]||

|-

| Number of abelian groups || 5|| (number of abelian groups of order <math>2^4</math>) times (number of abelian groups of order <math>3^1</math>) = ([[number of unordered integer partitions]] of 4) times ([[number of unordered integer partitions]] of 1) = <math>5 \times 1 = 5</math>. 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]]) = <math>14 \times 1 = 14</math>. 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|| {{only two prime factors hence solvable}}

|-

| Number of simple groups || 0|| Follows from all groups of this order being solvable.

|}

| [[elementary abelian group:E16]] || 14 || 4 || || 49, 50, 51, 52

|}

==GAP implementation==

{{this order in GAP|order = 48|idgroup = yes}}

<pre>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.</pre>

Retrieved from "https://groupprops.subwiki.org/wiki/Special:MobileDiff/30915"