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Groups of order 48

1,868 bytes added, 17:43, 15 June 2011
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==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>
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