View source for Linear representation theory of groups of order 48

{{all given order groups-specific information|
order = 48|
information type = linear representation theory|
connective = of}}

==Degrees of irreducible representations==

{{irrep degrees facts to check against}}

===Grouping by degrees of irreducible representations===

{| class="sortable" border="1"
! Number of irreps of degree 1 !! Number of irreps of degree 2 !! Number of irreps of degree 3 !! Number of irreps of degree 4 !! Total number of irreducible representations = [[number of conjugacy classes]] !! Number of groups with these irreps !! Nilpotency class(es) attained !! Derived lengths attained !! Description of groups !! List of groups !! List of GAP IDs (second part)
|-
| 48 || 0 || 0 || 0 || 48 || 5 || 1 || 1 || all the [[abelian group]]s || || 2, 20, 23, 44, 52
|-
| 24 || 6 || 0 || 0 || 30 || 6 || 2 || 2 || all the non-abelian groups of nilpotency class two, obtained as direct products of such groups of order 16 and [[cyclic group:Z3]] || || 21, 22, 24, 45, 46, 47
|-
| 16 || 8 || 0 || 0 || 24 || 7 || not nilpotent || 2 || || ||1, 4, 9, 11, 35, 42, 51
|-
| 12 || 9 || 0 || 0 || 21 || 3 || 3 || 2 || all the groups of nilpotency class three, obtained as direct products of such groups of order 16 with [[cyclic group:Z3]] || [[direct product of D16 and Z3]], [[direct product of SD16 and Z3]], [[direct product of Q16 and Z3]] || 25, 26, 27
|-
| 12 || 0 || 4 || 0 || 16 || 2 || not nilpotent || 2 || || || 31, 49
|-
| 8 || 10 || 0 || 0 || 18 || 10 || not nilpotent || 2 || || || 5, 10, 12, 13, 14, 19, 34, 36, 37, 43
|-
| 8 || 6 || 0 || 1 || 15 || 4 || not nilpotent || 2 || || || 38, 39, 40, 41
|-
| 6 || 6 || 2 || 0 || 14 || 2 || not nilpotent || 3 || || || 32, 33
|-
| 4 || 11 || 0 || 0 || 15 || 3 || not nilpotent || 2 || || || 6, 7, 8
|-
| 4 || 7 || 0 || 1 || 12 || 4 || not nilpotent || 2 || || || 15, 16, 17, 18
|-
| 4 || 2 || 4 || 0 || 10 || 2 || not nilpotent || 3 || || || 30, 48
|-
| 3 || 0 || 5 || 0 || 8 || 2 || not nilpotent || 2 || || || 3, 50
|-
| 2 || 3 || 2 || 1 || 8 || 2 || not nilpotent || 4 || all the derived length four groups || [[general linear group:GL(2,3)]] and [[binary octahedral group]] || 28, 29
|}

The GAP code to construct this is as follows:<toggledisplay>

We use the coded (''not in-built'') function [[GAP:IrrepDegreeGroupingFull|IrrepDegreeGroupingFull]] (follow link to get code) as follows:

<pre>gap>IrrepDegreeGroupingFull(48);
[ [ [ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ], [ 28, 29 ] ],
  [ [ [ 1, 3 ], [ 3, 5 ] ], [ 3, 50 ] ],
  [ [ [ 1, 4 ], [ 2, 2 ], [ 3, 4 ] ], [ 30, 48 ] ],
  [ [ [ 1, 4 ], [ 2, 7 ], [ 4, 1 ] ], [ 15, 16, 17, 18 ] ],
  [ [ [ 1, 4 ], [ 2, 11 ] ], [ 6, 7, 8 ] ],
  [ [ [ 1, 6 ], [ 2, 6 ], [ 3, 2 ] ], [ 32, 33 ] ],
  [ [ [ 1, 8 ], [ 2, 6 ], [ 4, 1 ] ], [ 38, 39, 40, 41 ] ],
  [ [ [ 1, 8 ], [ 2, 10 ] ], [ 5, 10, 12, 13, 14, 19, 34, 36, 37, 43 ] ],
  [ [ [ 1, 12 ], [ 2, 9 ] ], [ 25, 26, 27 ] ],
  [ [ [ 1, 12 ], [ 3, 4 ] ], [ 31, 49 ] ],
  [ [ [ 1, 16 ], [ 2, 8 ] ], [ 1, 4, 9, 11, 35, 42, 51 ] ],
  [ [ [ 1, 24 ], [ 2, 6 ] ], [ 21, 22, 24, 45, 46, 47 ] ],
  [ [ [ 1, 48 ] ], [ 2, 20, 23, 44, 52 ] ] ]</pre></toggledisplay>

==Relation with other orders==

===Divisors of the order===

{| class="sortable" border="1"
! Divisor !! Quotient value !! Information on linear representation theory !! Relationship (subgroup perspective) !! Relationship (quotient perspective)
|-
| 2 || 24 || [[linear representation theory of cyclic group:Z2]] || {{fillin}} || {{fillin}}
|-
| 3 || 16 || [[linear representation theory of cyclic group:Z3]] || {{fillin}} || {{fillin}}
|-
| 4 || 12 || [[linear representation theory of groups of order 4]] || {{fillin}} || {{fillin}}
|-
| 6 || 8 || [[linear representation theory of groups of order 6]] || {{fillin}} || {{fillin}}
|-
| 8 || 6 || [[linear representation theory of groups of order 8]] || {{fillin}} || {{fillin}}
|-
| 12 || 4 || [[linear representation theory of groups of order 12]] || {{fillin}} || {{fillin}}
|-
| 16 || 3 || [[linear representation theory of groups of order 16]] || {{fillin}} || {{fillin}}
|-
| 24 || 2 || [[linear representation theory of groups of order 24]] || {{fillin}} || {{fillin}}
|}

===Multiples of the order===

{| class="sortable" border="1"
! Multiplier (other factor) !! Multiple !! Information on linear representation theory !! Relationship (subgroup perspective) !! Relationship (quotient perspective)
|-
| 2 || 96 || [[linear representation theory of groups of order 96]] || ||
|-
| 3 || 144 || [[linear representation theory of groups of order 144]] || ||
|-
| 4 || 192 || [[linear representation theory of groups of order 192]] || ||
|-
| 5 || 240 || [[linear representation theory of groups of order 240]] || ||
|-
| 6 || 288 || [[linear representation theory of groups of order 288]] || ||
|-
| 7 || 336 || [[linear representation theory of groups of order 336]] || ||
|-
| 8 || 384 || [[linear representation theory of groups of order 384]] || ||
|-
| 9 || 432 || [[linear representation theory of groups of order 432]] || ||
|-
| 10 || 480 || [[linear representation theory of groups of order 480]] || ||
|}