Table 1.

Characteristics of fullerenes with 20 ≤ n ≤ 84 vertices

									IDs of probable cages
n	Faces	Hexs	Graphically possible Cages	IPR cages	Lowest Np for each n	Number of lowest Np isomers	IDs of isomers with lowest Np	Number of probable cages	FM	Schonflies
20-60			5770	1		35		16
20-70			30,579	2		56		17
60-84			218,551	51		71		52
20-84			222,509	51		105		66
20	12	0	1	0	30	1	1	1	1	Ih
24	14	2	1	0	24	1	1	1	1	D6d
26	15	3	1	0	21	1	1	1	1	D3h
28	16	4	2	0	18	1	2	1	2	Td
30	17	5	3	0	17	1	3	0
32	18	6	6	0	15	1	6	1	6	D3
34	19	7	6	0	14	1	5	0
36	20	8	15	0	12	2	14, 15	2	14, 15	D2d(II), D6h
38	21	9	17	0	11	1	17	1	17	C2(V)
40	22	10	40	0	10	2	38, 39	2	38, 39	D2(III), D5d(II)
42	23	11	45	0	9	1	45	1	45	D3
44	24	12	89	0	8	2	75, 89	2	75, 89	D2(III), D2(VI)
46	25	13	116	0	8	7	99,103,107,108,109,114,116	0
48	26	14	199	0	7	4	171, 196, 197, 199	0
50	27	15	271	0	5	1	271	1	271	D5h(II)
52	28	16	437	0	5	1	422	0
54	29	17	580	0	4	1	540	0
56	30	18	924	0	4	4	843, 864, 913, 916	0
58	31	19	1205	0	3	1	1205	0
60	32	20	1812	1	0	1	1812 (IPR-1)	2	1784, 1812 (IPR-1)	D6h(II), Ih
62	33	21	2385	0	3	3	2184, 2377, 2378	0
64	34	22	3465	0	2	3	3451, 3452, 2378	0
66	35	23	4478	0	2	3	4169, 4348, 4466	0
68	36	24	6332	0	2	11	6073,6094,6146,6148,6149,6195, 6198,6269,6270,6290,6328	0
70	37	25	8149	1	0	1	All IPR	1	All IPR
72	38	26	11190	1	0	1	All IPR	1	All IPR
74	39	27	14246	1	0	1	All IPR	1	All IPR
76	40	28	19151	2	0	2	All IPR	2	All IPR
78	41	29	24109	5	0	5	All IPR	5	All IPR
80	42	30	31924	7	0	7	All IPR	7	All IPR
82	43	31	39718	9	0	9	All IPR	9	All IPR
84	44	32	51592	24	0	24	All IPR	24	All IPR
86	45	33	63761	19	0	19	All IPR
88	46	34	81738	35	0	35	All IPR
90	47	35	99918	46	0	46	All IPR
92	48	36	126409	86	0	86	All IPR
94	49	37	153493	134	0	134	All IPR
96	50	38	191839	187	0	187	All IPR
98	51	39	231017	259	0	259	All IPR
100	52	40	285913	450	0	450	All IPR

The number of hexagons (hexs) is (n - 20)/2. With 12 pentagons, the number of faces is 12 more. The number of graphically possible cages and graphically possible cages that obey the isolated pentagon rule (IPR cages) were counted by use of the Fullgen program or the CaGe program, which use a productive algorithm (38). For each n, the isomer with the lowest number of pentagon pairs (N_p) is listed according to the (FM) isomer numbering in Fowler and Manolopoulos (24). These lowest N_p values are listed in Table 4.1 of Fowler and Manolopoulos. Probable cages have no improbable Rings, and the number and identify of these are listed for each n. Cages with 84 < n ≤ 100 were not investigated with respect to improbable rings.