Ambivalent but not rational groups

Groups whose character table is real-valued but not integer-valued.

order	group	conjugacy classes	class signature
10	D₅ #17	4	2x2:5,5:2
14	D₇ #26	5	3x2:7,7:2
16	D₈ #35	7	1:2,2:4,2x2:8,2x4:2
16	Q₁₆ #37	7	1:2,2:4,2x2:8,2x4:4
18	D₉ #44	6	2:3,3x2:9,9:2
20	C₂×D₅ #53	8	1:2,2x2:5,2x2:10,2x5:2
22	D₁₁ #57	7	5x2:11,11:2
24	Dic₆ #63	9	1:2,2:3,2:4,2:6,2x2:12,2x6:4
24	D₁₂ #65	9	1:2,2:3,2:4,2:6,2x2:12,2x6:2
26	D₁₃ #77	8	6x2:13,13:2
28	C₂×D₇ #86	10	1:2,3x2:7,3x2:14,2x7:2
30	D₁₅ #91	9	2:3,2x2:5,4x2:15,15:2
32	D₁₆ #111	11	1:2,2:4,2x2:8,4x2:16,2x8:2
32	Q₃₂ #113	11	1:2,2:4,2x2:8,4x2:16,2x8:4
32	C₂×D₈ #132	14	3x1:2,2x2:4,4x2:8,4x4:2
32	C₂×Q₁₆ #134	14	3x1:2,2x2:4,4x2:8,4x4:4
34	D₁₇ #146	10	8x2:17,17:2
36	C₂×D₉ #152	12	1:2,2:3,2:6,3x2:9,3x2:18,2x9:2
38	D₁₉ #164	11	9x2:19,19:2
40	Dic₁₀ #171	13	1:2,2:4,2x2:5,2x2:10,4x2:20,2x10:4
40	D₂₀ #173	13	1:2,2:4,2x2:5,2x2:10,4x2:20,2x10:2
40	C₂²×D₅ #180	16	3x1:2,2x2:5,6x2:10,4x5:2
42	D₂₁ #187	12	2:3,3x2:7,6x2:21,21:2
44	C₂×D₁₁ #192	14	1:2,5x2:11,5x2:22,2x11:2
46	D₂₃ #196	13	11x2:23,23:2
48	D₂₄ #205	15	1:2,2:3,2:4,2:6,2x2:8,2x2:12,4x2:24,2x12:2
48	Dic₁₂ #206	15	1:2,2:3,2:4,2:6,2x2:8,2x2:12,4x2:24,2x12:4
48	#226	8	1:2,6:4,2x6:8,8:3,8:6,12:4
48	C₂×Dic₆ #232	18	3x1:2,2:3,2x2:4,3x2:6,4x2:12,4x6:4
48	C₂×D₁₂ #234	18	3x1:2,2:3,2x2:4,3x2:6,4x2:12,4x6:2
50	D₂₅ #253	14	2x2:5,10x2:25,25:2
50	C₅⋊D₅ #256	14	12x2:5,25:2
52	C₂×D₁₃ #262	16	1:2,6x2:13,6x2:26,2x13:2
54	D₂₇ #265	15	2:3,3x2:9,9x2:27,27:2
54	C₃⋊D₉ #271	15	4x2:3,9x2:9,27:2
56	Dic₁₄ #284	17	1:2,2:4,3x2:7,3x2:14,6x2:28,2x14:4
56	D₂₈ #286	17	1:2,2:4,3x2:7,3x2:14,6x2:28,2x14:2
56	C₂²×D₇ #293	20	3x1:2,3x2:7,9x2:14,4x7:2
58	D₂₉ #297	16	14x2:29,29:2
60	A₅ #304	5	2x12:5,15:2,20:3
60	D₃×D₅ #307	12	2:3,2x2:5,3:2,2x4:15,5:2,2x6:10,10:6,15:2
60	C₂×D₁₅ #311	18	1:2,2:3,2x2:5,2:6,2x2:10,4x2:15,4x2:30,2x15:2
62	D₃₁ #314	17	15x2:31,31:2
64	D₃₂ #371	19	1:2,2:4,2x2:8,4x2:16,8x2:32,2x16:2
64	Q₆₄ #373	19	1:2,2:4,2x2:8,4x2:16,8x2:32,2x16:4
64	#447	19	3x1:2,2x2:2,2x2:4,4x4:2,4:4,4x4:8,8:2,8:4
64	#451	19	3x1:2,2x2:2,2x2:4,5x4:4,4x4:8,2x8:4
64	#493	22	3x1:2,6x2:4,8x2:8,4x8:2
64	#494	22	3x1:2,6x2:4,8x2:8,4x8:4
64	#500	22	3x1:2,6x2:4,8x2:8,4x8:4
64	C₂×D₁₆ #505	22	3x1:2,2x2:4,4x2:8,8x2:16,4x8:2
64	C₂×Q₃₂ #507	22	3x1:2,2x2:4,4x2:8,8x2:16,4x8:4
64	#509	16	1:2,2:2,2x2:4,2x2:8,4:8,4x4:16,3x8:2,8:4
64	#510	16	1:2,2:2,2x2:4,2x2:8,4:8,4x4:16,8:2,3x8:4
64	C₂²×D₈ #569	28	7x1:2,4x2:4,8x2:8,8x4:2
64	C₂²×Q₁₆ #571	28	7x1:2,4x2:4,8x2:8,8x4:4
64	#576	22	1:2,3x2:2,4x2:4,2x2:8,6x4:2,2x4:4,3x4:8
64	#578	22	1:2,3x2:2,4x2:4,2x2:8,2x4:2,6x4:4,3x4:8
66	D₃₃ #590	18	2:3,5x2:11,10x2:33,33:2
68	C₂×D₁₇ #596	20	1:2,8x2:17,8x2:34,2x17:2
70	D₃₅ #601	19	2x2:5,3x2:7,12x2:35,35:2
72	Dic₁₈ #607	21	1:2,2:3,2:4,2:6,3x2:9,2x2:12,3x2:18,6x2:36,2x18:4
72	D₃₆ #609	21	1:2,2:3,2:4,2:6,3x2:9,2x2:12,3x2:18,6x2:36,2x18:2
72	(C₂²⋊C₉)⋊C₂ #618	9	2:3,3:2,6:6,3x8:9,18:2,18:4
72	C₂²×D₉ #620	24	3x1:2,2:3,3x2:6,3x2:9,9x2:18,4x9:2
72	#627	15	1:2,2x2:3,2x2:6,4:3,4:6,2x6:4,4x6:12,18:4
72	#634	21	1:2,4x2:3,2:4,4x2:6,8x2:12,2x18:4
72	C₄⋊(C₃⋊D₃) #636	21	1:2,4x2:3,2:4,4x2:6,8x2:12,2x18:2
74	D₃₇ #655	20	18x2:37,37:2
76	C₂×D₁₉ #662	22	1:2,9x2:19,9x2:38,2x19:2
78	D₃₉ #669	21	2:3,6x2:13,12x2:39,39:2
80	D₄₀ #678	23	1:2,2:4,2x2:5,2x2:8,2x2:10,4x2:20,8x2:40,2x20:2
80	Dic₂₀ #679	23	1:2,2:4,2x2:5,2x2:8,2x2:10,4x2:20,8x2:40,2x20:4
80	C₂×Dic₁₀ #706	26	3x1:2,2x2:4,2x2:5,6x2:10,8x2:20,4x10:4
80	C₂×D₂₀ #708	26	3x1:2,2x2:4,2x2:5,6x2:10,8x2:20,4x10:2
80	D₄×D₅ #710	20	1:2,2x2:2,2:4,2x2:5,2x2:10,4x4:10,2x4:20,2x5:2,2x10:2,10:4
80	Q₈×D₅ #712	20	1:2,3x2:4,2x2:5,2x2:10,6x4:20,2x5:2,3x10:4
80	C₂³×D₅ #722	32	7x1:2,2x2:5,14x2:10,8x5:2
82	D₄₁ #739	22	20x2:41,41:2
84	D₃×D₇ #749	15	2:3,3x2:7,3:2,3x4:21,3x6:14,7:2,14:6,21:2
84	C₂×D₂₁ #755	24	1:2,2:3,2:6,3x2:7,3x2:14,6x2:21,6x2:42,2x21:2
86	D₄₃ #758	23	21x2:43,43:2
88	Dic₂₂ #763	25	1:2,2:4,5x2:11,5x2:22,10x2:44,2x22:4
88	D₄₄ #765	25	1:2,2:4,5x2:11,5x2:22,10x2:44,2x22:2
88	C₂²×D₁₁ #771	28	3x1:2,5x2:11,15x2:22,4x11:2
90	D₄₅ #776	24	2:3,2x2:5,3x2:9,4x2:15,12x2:45,45:2
90	C₃⋊D₁₅ #782	24	4x2:3,2x2:5,16x2:15,45:2
92	C₂×D₂₃ #787	26	1:2,11x2:23,11x2:46,2x23:2
94	D₄₇ #791	25	23x2:47,47:2
96	D₄₈ #799	27	1:2,2:3,2:4,2:6,2x2:8,2x2:12,4x2:16,4x2:24,8x2:48,2x24:2
96	Dic₂₄ #801	27	1:2,2:3,2:4,2:6,2x2:8,2x2:12,4x2:16,4x2:24,8x2:48,2x24:4
96	#869	30	3x1:2,2:3,6x2:4,3x2:6,12x2:12,4x12:4
96	#874	30	3x1:2,2:3,6x2:4,3x2:6,12x2:12,4x12:2
96	#882	24	3x1:2,2x2:2,2:3,3x2:6,2x4:4,2x4:6,4x4:12,4x6:2,12:2,12:4
96	#888	24	3x1:2,2:3,2x2:4,3x2:6,2x4:4,6x4:12,4x6:4,2x12:4
96	C₂×D₂₄ #903	30	3x1:2,2:3,2x2:4,3x2:6,4x2:8,4x2:12,8x2:24,4x12:2
96	C₂×Dic₁₂ #905	30	3x1:2,2:3,2x2:4,3x2:6,4x2:8,4x2:12,8x2:24,4x12:4
96	#908	21	1:2,2:2,2:3,2x2:4,2:6,2x2:12,4:6,2x4:8,4:12,4x4:24,3x12:2,12:4
96	#909	21	1:2,2:2,2:3,2x2:4,2:6,2x2:12,4:6,2x4:8,4:12,4x4:24,12:2,3x12:4
96	D₃×D₈ #910	21	1:2,2:3,2:4,2:6,2x2:8,2x3:2,2x4:2,4:12,2x4:24,6:4,2x6:8,2x8:6,2x12:2

About ambivalent but not rational groups →