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.
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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

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	${\mathrm{\Gamma }}_{2}h$	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	${\mathrm{\Gamma }}_{2}i$	2	16	4	0	20
SmallGroup(32,5)	5	20	${\mathrm{\Gamma }}_2{j}_1$	2	16	4	0	20
Faithful semidirect product of E8 and Z4	6	46	${\mathrm{\Gamma }}_7{a}_1$	3	8	2	1	11
SmallGroup(32,7)	7	47	${\mathrm{\Gamma }}_7{a}_2$	3	8	2	1	11
SmallGroup(32,8)	8	48	${\mathrm{\Gamma }}_7{a}_3$	3	8	2	1	11
SmallGroup(32,9)	9	27	${\mathrm{\Gamma }}_3{c}_1$	3	8	6	0	14
SmallGroup(32,10)	10	28	${\mathrm{\Gamma }}_3{c}_2$	3	8	6	0	14
Wreath product of Z4 and Z2	11	31	${\mathrm{\Gamma }}_{3}e$	3	8	6	0	14
Nontrivial semidirect product of Z4 and Z8	12	21	${\mathrm{\Gamma }}_2{j}_2$	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	${\mathrm{\Gamma }}_{3}f$	3	8	6	0	14
Direct product of Z16 and Z2	16	6	$(41)$	1	32	0	0	32
M32	17	22	${\mathrm{\Gamma }}_{2}k$	2	16	4	0	20
Dihedral group:D32	18	49	${\mathrm{\Gamma }}_8{a}_1$	4	4	7	0	11
Semidihedral group:SD32	19	50	${\mathrm{\Gamma }}_8{a}_2$	4	4	7	0	11
Generalized quaternion group:Q32	20	51	${\mathrm{\Gamma }}_8{a}_3$	4	4	7	0	11
Direct product of Z4 and Z4 and Z2	21	3	$(2^{2}1)$	1	32	0	0	32
Direct product of SmallGroup(16,3) and Z2	22	11	${\mathrm{\Gamma }}_2{c}_1$	2	16	4	0	20
Direct product of SmallGroup(16,4) and Z2	23	12	${\mathrm{\Gamma }}_2{c}_2$	2	16	4	0	20
SmallGroup(32,24)	24	16	${\mathrm{\Gamma }}_{2}f$	2	16	4	0	20
Direct product of D8 and Z4	25	14	${\mathrm{\Gamma }}_2{e}_1$	2	16	4	0	20
Direct product of Q8 and Z4	26	15	${\mathrm{\Gamma }}_2{e}_2$	2	16	4	0	20
SmallGroup(32,27)	27	33	${\mathrm{\Gamma }}_4{a}_1$	2	8	6	0	14
SmallGroup(32,28)	28	36	${\mathrm{\Gamma }}_4{b}_1$	2	8	6	0	14
SmallGroup(32,29)	29	37	${\mathrm{\Gamma }}_4{b}_2$	2	8	6	0	14
SmallGroup(32,30)	30	38	${\mathrm{\Gamma }}_4{c}_1$	2	8	6	0	14
SmallGroup(32,31)	31	39	${\mathrm{\Gamma }}_4{c}_2$	2	8	6	0	14
SmallGroup(32,32)	32	40	${\mathrm{\Gamma }}_4{c}_3$	2	8	6	0	14
SmallGroup(32,33)	33	41	${\mathrm{\Gamma }}_{4}d$	2	8	6	0	14
Generalized dihedral group for direct product of Z4 and Z4	34	34	${\mathrm{\Gamma }}_4{a}_2$	2	8	6	0	14
SmallGroup(32,35)	35	35	${\mathrm{\Gamma }}_4{a}_3$	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	${\mathrm{\Gamma }}_{2}d$	2	16	4	0	20
Central product of D8 and Z8	38	17	${\mathrm{\Gamma }}_{2}g$	2	16	4	0	20
Direct product of D16 and Z2	39	23	${\mathrm{\Gamma }}_3{a}_1$	3	8	6	0	14
Direct product of SD16 and Z2	40	24	${\mathrm{\Gamma }}_3{a}_2$	3	8	6	0	14
Direct product of Q16 and Z2	41	25	${\mathrm{\Gamma }}_3{a}_3$	3	8	6	0	14
Central product of D16 and Z4	42	26	${\mathrm{\Gamma }}_{3}b$	3	8	6	0	14
Holomorph of Z8	43	44	${\mathrm{\Gamma }}_6{a}_1$	3	8	2	1	11
SmallGroup(32,44)	44	45	${\mathrm{\Gamma }}_6{a}_2$	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	${\mathrm{\Gamma }}_2{a}_1$	2	16	4	0	20
Direct product of Q8 and V4	47	9	${\mathrm{\Gamma }}_2{a}_2$	2	16	4	0	20
Direct product of SmallGroup(16,13) and Z2	48	10	${\mathrm{\Gamma }}_{2}b$	2	16	4	0	20
Inner holomorph of D8	49	42	${\mathrm{\Gamma }}_5{a}_1$	2	16	0	1	17
Central product of D8 and Q8	50	43	${\mathrm{\Gamma }}_5{a}_2$	2	16	0	1	17
Elementary abelian group:E32	51	1	$(1^5)$	1	32	0	0	32

Here now is a grouping by degrees of irreducible representations:

Number of irreps of degree 1	Number of irreps of degree 2	Number of irreps of degree 4	Total number of irreps	Total number of groups	Nilpotency class(es) attained by these	Description of groups	List of groups	List of GAP IDs (ascending order)	List of Hall-Senior numbers (ascending order)	List of Hall-Senior symbols/families
32	0	0	32	7	1	all the abelian groups of order 32	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	1, 3, 16, 21, 36, 45, 51	1--7	${\mathrm{\Gamma }}_1$ (abelian)
16	4	0	20	15	2	inner automorphism group is Klein four-group and derived subgroup is 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.	2, 4, 5, 12, 17, 22, 23, 24, 25, 26, 37, 38, 46, 47, 48	8--22	${\mathrm{\Gamma }}_2$
16	0	1	17	2	2	the two extraspecial groups	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43	${\mathrm{\Gamma }}_5$ (the Hall-Senior family for extraspecial groups)
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.	9, 10, 11, 13, 14, 15, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 41, 42	23--41	${\mathrm{\Gamma }}_3$ (class three, ten groups numbers 23--32) and ${\mathrm{\Gamma }}_4$ (class two, nine groups, numbers 33--41)
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	${\mathrm{\Gamma }}_6$ and ${\mathrm{\Gamma }}_7$
4	7	0	11	3	4	the maximal class groups	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18, 19, 20	49--51	${\mathrm{\Gamma }}_8$