Linear representation theory of groups of order 32

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 32.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 32

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

Full listing

Here are the degrees of irreducible representations for all groups of order 32:

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Group	GAP ID second part	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Number of irreps of degree 1	Number of irreps of degree 2	Number of irreps of degree 4	Total number of irreps
Cyclic group:Z32	1	7	$(5)$	1	32	0	0	32
SmallGroup(32,2)	2	18	$\Gamma_2h$	2	16	4	0	20
Direct product of Z8 and Z4	3	5	$(32)$	1	32	0	0	20
Semidirect product of Z8 and Z4 of M-type	4	19	$\Gamma_2i$	2	16	4	0	20
SmallGroup(32,5)	5	20	$\Gamma_2j_1$	2	16	4	0	20
Faithful semidirect product of E8 and Z4	6	46	$\Gamma_7a_1$	3	8	2	1	11
SmallGroup(32,7)	7	47	$\Gamma_7a_2$	3	8	2	1	11
SmallGroup(32,8)	8	48	$\Gamma_7a_3$	3	8	2	1	11
SmallGroup(32,9)	9	27	$\Gamma_3c_1$	3	8	6	0	14
SmallGroup(32,10)	10	28	$\Gamma_3c_2$	3	8	6	0	14
Wreath product of Z4 and Z2	11	31	$\Gamma_3e$	3	8	6	0	14
Nontrivial semidirect product of Z4 and Z8	12	21	$\Gamma_2j_2$	2	16	4	0	20
Semidirect product of Z8 and Z4 of semidihedral type	13			3	8	6	0	14
Semidirect product of Z8 and Z4 of dihedral type	14			3	8	6	0	14
SmallGroup(32,15)	15	32	$\Gamma_3f$	3	8	6	0	14
Direct product of Z16 and Z2	16	6	$(41)$	1	32	0	0	32
M32	17	22	$\Gamma_2k$	2	16	4	0	20
Dihedral group:D32	18	49	$\Gamma_8a_1$	4	4	7	0	11
Semidihedral group:SD32	19	50	$\Gamma_8a_2$	4	4	7	0	11
Generalized quaternion group:Q32	20	51	$\Gamma_8a_3$	4	4	7	0	11
Direct product of Z4 and Z4 and Z2	21	3	$(2^21)$	1	32	0	0	32
Direct product of SmallGroup(16,3) and Z2	22	11	$\Gamma_2c_1$	2	16	4	0	20
Direct product of SmallGroup(16,4) and Z2	23	12	$\Gamma_2c_2$	2	16	4	0	20
SmallGroup(32,24)	24	16	$\Gamma_2f$	2	16	4	0	20
Direct product of D8 and Z4	25	14	$\Gamma_2e_1$	2	16	4	0	20
Direct product of Q8 and Z4	26	15	$\Gamma_2e_2$	2	16	4	0	20
SmallGroup(32,27)	27	33	$\Gamma_4a_1$	2	8	6	0	14
SmallGroup(32,28)	28	36	$\Gamma_4b_1$	2	8	6	0	14
SmallGroup(32,29)	29	37	$\Gamma_4b_2$	2	8	6	0	14
SmallGroup(32,30)	30	38	$\Gamma_4c_1$	2	8	6	0	14
SmallGroup(32,31)	31	39	$\Gamma_4c_2$	2	8	6	0	14
SmallGroup(32,32)	32	40	$\Gamma_4c_3$	2	8	6	0	14
SmallGroup(32,33)	33	41	$\Gamma_4d$	2	8	6	0	14
Generalized dihedral group for direct product of Z4 and Z4	34	34	$\Gamma_4a_2$	2	8	6	0	14
SmallGroup(32,35)	35	35	$\Gamma_4a_3$	2	8	6	0	14
Direct product of Z8 and V4	36	4	$(31^2)$	1	32	0	0	32
Direct product of M16 and Z2	37	13	$\Gamma_2d$	2	16	4	0	20
Central product of D8 and Z8	38	17	$\Gamma_2g$	2	16	4	0	20
Direct product of D16 and Z2	39	23	$\Gamma_3a_1$	3	8	6	0	14
Direct product of SD16 and Z2	40	24	$\Gamma_3a_2$	3	8	6	0	14
Direct product of Q16 and Z2	41	25	$\Gamma_3a_3$	3	8	6	0	14
Central product of D16 and Z4	42	26	$\Gamma_3b$	3	8	6	0	14
Holomorph of Z8	43	44	$\Gamma_6a_1$	3	8	2	1	11
SmallGroup(32,44)	44	45	$\Gamma_6a_2$	3	8	2	1	11
Direct product of E8 and Z4	45	2	$(21^3)$	1	32	0	0	32
Direct product of D8 and V4	46	8	$\Gamma_2a_1$	2	16	4	0	20
Direct product of Q8 and V4	47	9	$\Gamma_2a_2$	2	16	4	0	20
Direct product of SmallGroup(16,13) and Z2	48	10	$\Gamma_2b$	2	16	4	0	20
Inner holomorph of D8	49	42	$\Gamma_5a_1$	2	16	0	1	17
Central product of D8 and Q8	50	43	$\Gamma_5a_2$	2	16	0	1	17
Elementary abelian group:E32	51	1	$(1^5)$	1	32	0	0	32

Grouping by Hall-Senior families

Note that isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the same order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations. However, the same multiset of degrees of irreducible representations could be attained by more than one Hall-Senior family, as we see in the table below.

For the first three Hall-Senior families $\Gamma_1,\Gamma_2,\Gamma_3$, there are isoclinic groups of smaller order, hence the degrees of irreducible representations can be computed by first computing the degrees of irreducible representations of those isoclinic groups of smaller order and then scaling up the proportions based on the order. For instance, dihedral group:D8 of order 8 and family $\Gamma_2$ has 4 irreps of degree 1 and 1 of degree 2, so the groups in family $\Gamma_2$ and of order 32 have $4 \times (32/8) = 16$ irreps of degree 1 and $1 \times (32/8) = 4$ irreps of degree 2.

For more background on the Hall-Senior families business, see Groups of order 32#Families and classification.

Family name	Isomorphism class of inner automorphism group	Isomorphism class of derived subgroup	Number of members	Nilpotency class	Members	Second part of GAP ID of members (sorted ascending)	Hall-Senior numbers of members (sorted ascending)	Number of irreps of degree 1 (= order of abelianization)	Number of irreps of degree 2	Number of irreps of degree 4	Total number of irreps	Smallest order of group isoclinic to it	Degrees of irreps of smallest order isoclinic stem group
$\Gamma_1$	trivial group	trivial group	7	1	all abelian groups of order 32: [SHOW MORE] cyclic group:Z32, direct product of Z8 and Z4, direct product of Z16 and Z2, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of Z4 and E8, elementary abelian group:E32	[SHOW MORE] 1,3,16,21,36,45,51	1-7	32	0	0	32	1	1
$\Gamma_2$	Klein four-group	cyclic group:Z2	15	2	[SHOW MORE] direct product of D8 and V4, direct product of Q8 and V4, direct product of SmallGroup(16,13) and Z2, direct product of SmallGroup(16,3) and Z2, direct product of SmallGroup(16,4) and Z2, direct product of M16 and Z2, direct product of D8 and Z4, direct product of Q8 and Z4, SmallGroup(32,24), central product of D8 and Z8, SmallGroup(32,2), SmallGroup(32,5), SmallGroup(32,12), SmallGroup(32,12), M32, semidirect product of Z8 and Z4 of M-type	[SHOW MORE] 2,4,5,12,17,22,23,24,25,26,37,38,46,47,48	8-22	16	4	0	20	8	1 (4 times), 2 (1 time)
$\Gamma_3$	dihedral group:D8	cyclic group:Z4	10	3	[SHOW MORE] direct product of D16 and Z2, direct product of SD16 and Z2, direct product of Q16 and Z2, central product of D16 and Z4, semidirect product of Z8 and Z4 of dihedral type, semidirect product of Z8 and Z4 of semidihedral type, SmallGroup(32,9), SmallGroup(32,10), wreath product of Z4 and Z2, SmallGroup(32,15)	[SHOW MORE] 9,10,11,13,14,15,39,40,41,42	23-32	8	6	0	14	16	1 (4 times), 2 (3 times)
$\Gamma_4$	elementary abelian group:E8	Klein four-group	9	2	[SHOW MORE] SmallGroup(32,27), SmallGroup(32,28), SmallGroup(32,29), SmallGroup(32,30), SmallGroup(32,31), SmallGroup(32,32), SmallGroup(32,33), generalized dihedral group for direct product of Z4 and Z4, SmallGroup(32,35)	27-35	33-41	8	6	0	14	32	1 (8 times), 2 (6 times)
$\Gamma_5$	elementary abelian group:E16	cyclic group:Z2	2	2	extraspecial groups: [SHOW MORE] inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43	16	0	1	17	32	1 (16 times), 4 (1 time)
$\Gamma_6$	direct product of D8 and Z2	cyclic group:Z4	2	3	[SHOW MORE] holomorph of Z8, SmallGroup(32,44)	43,44	44,45	8	2	1	11	32	1 (8 times), 2 (2 times), 4 (1 time)
$\Gamma_7$	SmallGroup(16,3)	Klein four-group	3	3	faithful semidirect product of E8 and Z4, SmallGroup(32,7), SmallGroup(32,8)	6-8	46-48	8	2	1	11	32	1 (8 times), 2 (2 times), 4 (1 time)
$\Gamma_8$	dihedral group:D16	cyclic group:Z8	3	4	maximal class groups: [SHOW MORE] dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18-20	49-51	4	7	0	11	32	1 (4 times), 2 (7 times)

Grouping by degrees of irreducible representations

Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the total number of irreducible representations, which equals the number of conjugacy classes, is congruent to 32 mod 3, and hence congruent to 2 mod 3.

As noted above, isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the same order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations. However, the same multiset of degrees of irreducible representations could be attained by more than one Hall-Senior family, as we see in the table below.

For more background on the Hall-Senior families business, see Groups of order 32#Families and classification.

For more on the subgroup structure stuff (the two right-most columns) see subgroup structure of groups of order 32#Abelian subgroups.

Number of irreps of degree 1 (= order of abelianization)	Number of irreps of degree 2	Number of irreps of degree 4	Total number of irreps (= number of conjugacy classes)	Total number of groups	Nilpotency class(es) attained by these	List of groups	List of GAP IDs (ascending order)	List of Hall-Senior numbers (ascending order)	List of Hall-Senior families (equivalence classes under isoclinism)	Order of inner automorphism group (bounds square of degree of irreducible representation)	Minimum possible index of abelian normal subgroup (degree of irreducible representation divides index of abelian normal subgroup)
32	0	0	32	7	1	[SHOW MORE] cyclic group:Z32, direct product of Z8 and Z4, direct product of Z16 and Z2, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of E8 and Z4, elementary abelian group:E32	[SHOW MORE] 1, 3, 16, 21, 36, 45, 51	1--7	$\Gamma_1$ (abelian groups)	1	1
16	4	0	20	15	2	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.	[SHOW MORE] 2, 4, 5, 12, 17, 22, 23, 24, 25, 26, 37, 38, 46, 47, 48	8--22	$\Gamma_2$	4	2
8	6	0	14	19	2,3	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.	[SHOW MORE] 9, 10, 11, 13, 14, 15, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 41, 42	23--41	$\Gamma_3$ and $\Gamma_4$	8	2
16	0	1	17	2	2	[SHOW MORE] inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43	$\Gamma_5$ (extraspecial groups)	16	4
8	2	1	11	5	3	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, 7, 8, 43, 44	44--48	$\Gamma_6$ and $\Gamma_7$	16	4
4	7	0	11	3	4	[SHOW MORE] dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18, 19, 20	49--51	$\Gamma_8$ (maximal class groups)	16	2

Here is the GAP code to generate this information:

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We use the (not in-built, but coded) function IrrepDegreeGroupingFull (follow link to get code) to get the information on groups of order 32 grouped by their degrees of irreducible representations:

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

Correspondence between degrees of irreducible representations and conjugacy class sizes

See also element structure of groups of order 32#Conjugacy class sizes.

For groups of order 32, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Number of conjugacy classes of size 1	Number of conjugacy classes of size 2	Number of conjugacy classes of size 4	Number of conjugacy classes of size 8	Total number of conjugacy classes = number of irreducible representations	Number of irreps of degree 1	Number of irreps of degree 2	Number of irreps of degree 4
32	0	0	0	32	32	0	0
8	12	0	0	20	16	4	0
2	15	0	0	17	16	0	1
4	6	4	0	14	8	6	0
2	3	6	0	11	8	2	1
2	7	0	2	11	4	7	0

Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders:

• Degrees of irreducible representations need not determine conjugacy class size statistics
• Conjugacy class size statistics need not determine degrees of irreducible representations