Bol Loops of Order 16 with 9 Involutions

As part of my enumeration of the Bol loops of order 16 with nontrivial centre, here I list just the 508 loops which are non-associative with exactly nine involutions. Please see the parent page for notation, including my conventions for naming of loops and table entries. I would appreciate an email message () from you if you have any comments regarding this list.

One loop in this category has |Z(L)|=1, namely 16.9.8.3. The remaining 507 loops in this list all have |Z(L)|=2 and so the graph Comm(L) is defined. This graph is displayed as black-on-white if it has at most 10 edges; otherwise its complement (which has at most 10 edges) is displayed as white-on-black. In either case, isolated vertices are not displayed.

No. Comm(L) |C(L)|=2 |C(L)|=4 |C(L)|=6 |C(L)|=8
1empty graph 435      
2       1, 2
5 428, 430, 431      
6   8, 12, 38, 39    
9 95, 165, 191, 443      
10 0, 1, 4      
12 169, 289, 370, 373, 434      
13 3, 6      
15 299      
16 65, 159, 393      
18 7, 9      
19 16      
21 114, 156, 211, 276, 277, 291, 296      
23 64, 322, 374      
26 103, 323, 357, 438      
27 122, 219, 235, 260      
29 34, 52, 144, 337      
32 387, 410, 444      
33 2, 5      
35 321, 439      
38   2    
40 99, 116, 269, 270      
42 76, 281, 344      
44 13, 75, 87      
46 442      
47 139, 168, 202, 292      
51 88, 228, 250, 278, 317, 342, 406      
52 237, 309, 413      
55 79      
57 96, 318, 331, 335      
58   24, 29, 33, 47    
60   16, 18, 22, 28, 48, 49, 50    
63 194, 204, 372, 402      
65 59, 124, 412      
67 93, 118, 399, 446      
69 100, 134, 152, 161, 200, 221, 253, 343, 362, 388, 407      
71 146, 185, 212, 416      
73 18, 190, 230, 302      
76 94, 252, 271, 396      
78 220, 238, 320, 397      
79 101, 133, 189, 234, 279, 306, 368      
81 11, 61, 62, 197, 274, 285, 336, 384, 394, 408      
83       0
85 356, 436, 437      
86   6, 7, 10, 31, 32, 34, 36, 43    
89 131, 272, 346, 400      
91 199, 304, 334, 383      
93 42, 47      
94   0, 15, 37, 51, 54    
97 44, 49, 256, 264, 282      
99 115, 225, 258, 366      
101 148, 184, 188, 332, 350, 381, 389, 409      
102 145, 150, 155, 206, 227, 233, 363      
104 208, 347, 349, 380      
106 163, 177, 226, 386      
108 149, 192, 275, 295      
110 129, 130, 166, 377      
113 125, 218, 242, 341      
114     0, 1, 2  
116 37, 50      
118 14, 20, 153      
121 119, 157, 262, 267      
123 41, 48, 147, 440      
126 36, 43, 46      
128 23, 30, 348      
130 107, 205, 223, 244      
131 102, 109, 179, 361      
133 39, 53, 91, 259, 352      
135 154, 170, 187, 207      
137 143, 280, 310, 403      
140 113, 198, 245, 378      
142 175, 178, 325, 391      
144 266, 359, 441      
145 10, 35, 40, 51, 160, 209, 315      
148 80, 83, 432      
149   3    
151 70, 382      
153 217, 229, 243      
155 67, 419, 422, 429      
158 112, 128, 311      
160 141, 151, 248, 308      
162 97, 340, 398      
164 15, 57, 232, 312      
166 195, 214, 249      
168   1, 5, 13, 42, 53    
170 136, 142, 254, 290      
173 162, 255, 297, 415      
175 25, 33, 224, 364      
176   11, 17, 35    
180 69      
181 241, 316, 324, 330      
183 201, 246, 286, 345      
186 104, 193, 215, 263, 283, 284, 339, 353, 360, 369, 423      
188 38, 45, 54, 55, 56, 82, 236, 294, 300, 395, 424      
190 181, 319, 425      
192 28, 31, 77, 84, 132, 158, 307      
194 213, 293, 376, 417      
196 92, 126, 135, 265, 327, 333, 365, 414      
198 105, 108, 172, 303, 371, 418, 420      
199 110, 216, 328, 358      
201 58, 73, 89      
203   4    
205 182, 239, 268, 305      
207 117, 210, 251, 367      
209 127, 231, 385      
211 176, 183, 273, 390      
213 19, 66, 355      
214   9, 14, 40, 41    
217 24, 29, 78, 86      
220 222, 411, 421, 426, 433      
221   21, 23, 25, 30, 46    
223 12, 68, 71, 74, 164      
226 329, 351, 392      
228 27, 32, 60, 121, 140, 314, 427      
229 171, 186, 288, 298      
231 63, 72, 379      
234 21, 26, 137, 287      
236   19, 20, 26, 27, 44, 45, 52    
238 240, 247, 313, 404      
240 120, 174, 326, 405      
242 196, 301, 401, 445      
243 167      
245 17, 22, 81, 85, 111, 257, 354, 375      
247 123, 338      
250 90, 138, 180, 203      
252 98, 106, 173, 261      

/ revised November, 2001