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Groups of order 48

, 17:43, 15 June 2011
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==Statistics at a glance==

The number 48 has prime factorization $48 = 2^4 \cdot 3$. {{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 $2^4$) 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|| {{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>