Linear representation theory of groups of order 48

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 48.
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Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Grouping by degrees of irreducible representations

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 groups		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:[SHOW MORE]

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

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 ] ] ]

Relation with other orders

Divisors of the order

Divisor	Quotient value	Information on linear representation theory	Relationship (subgroup perspective)	Relationship (quotient perspective)
2	24	linear representation theory of cyclic group:Z2	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
3	16	linear representation theory of cyclic group:Z3	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
4	12	linear representation theory of groups of order 4	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
6	8	linear representation theory of groups of order 6	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
8	6	linear representation theory of groups of order 8	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
12	4	linear representation theory of groups of order 12	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
16	3	linear representation theory of groups of order 16	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
24	2	linear representation theory of groups of order 24	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Multiples of the order

Multiplier (other factor)	Multiple	Information on linear representation theory
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