Bol Loops of Order 16 with 7 Involutions

As part of my enumeration of the Bol loops of order 16 with nontrivial centre, here I list just the 532 loops which are non-associative with exactly seven 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.

7 loops in this category have |Z(L)|=1, namely

• 4 loops with |C(L)|=2:     16.7.2.441,   16.7.2.442,   16.7.2.443,   16.7.2.444;   and
• 2 loops with |C(L)|=4:     16.7.4.77,   16.7.4.78;   and
• 1 loop with |C(L)|=8:     16.7.8.3.

Also 17 loops were found in this category with |Z(L)|=4, namely

3 loops with |C(L)|=8:     16.7.8.0,   16.7.8.1,   16.7.8.2.

The remaining 508 loops found in this category 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
3		416, 421
4				0, 1, 5
7		229, 340, 370, 396
8		138, 147, 328, 380
11		9, 10, 73, 80, 263
14			47
17		202, 204, 397
20		3, 4, 5, 270, 436, 438
22		95, 160, 295, 310, 334, 351, 386
24		108, 142, 409
25			26, 39, 48, 65
28		102, 110, 212, 239
30		65, 85, 355, 361
31		52, 131, 150
34			29, 62
37		7, 439
39		48, 50, 158, 178
41		8
43		83, 215, 216
45			2, 3, 38, 66
48		283, 369, 395, 403
49			18, 19
50			24, 25, 43, 44, 50, 56, 73
53		43, 417, 422
54		161
56		126, 273, 284, 392
59		149, 232, 251, 345
61		192, 243, 279, 321, 330, 375, 431
62		174, 244, 296, 377
64		111, 249, 254
66		113, 132, 137, 382
68		11, 36, 94, 173, 188, 220, 246, 247, 248, 260, 399
72		148, 317, 339, 406
74		329, 344, 366, 418
75		193, 277, 327, 378
77		86, 141, 211, 294
80		123, 186, 222, 299, 300, 425, 430
82		256, 266, 292, 293, 336, 360, 373, 401, 407, 411
84			60, 68, 70
87		77, 112, 301, 348, 349, 393, 402, 419
88		61, 127, 227, 384
90		109, 119, 120, 176
92		106, 116
95		187, 311, 320
96		171, 252, 319, 323, 388
98		23, 28, 214, 307
100			31, 33, 37, 40, 53, 54, 55, 71
103		225, 278, 314, 324, 338, 426, 428
105		100, 189, 238, 371
107		53, 281, 288, 363
109		60, 180, 234, 313
111		22, 26, 303, 372
112		93, 153, 159, 200
115		27, 308, 408
117		114, 342
119		99, 208, 223
120			27, 41, 51, 63
122		144, 207, 250, 337
124		6
125		21, 331, 362
127		41, 78, 343
129		128, 163, 235, 276
132		29, 30, 233, 309
134		96, 118, 315, 335, 352
136		71, 136, 182, 367
138		105, 122, 175, 259
139		146, 157, 350, 413
141		67, 151, 196, 365
143			35, 59, 64
146		64, 166, 185, 206, 333, 390, 410
147			4, 5
149			16, 76
152		0, 433, 435, 440
154		91, 125, 432
156		1, 2, 217, 231, 347, 434, 437
157		39, 124, 304
159		143, 201, 253, 258
161			28, 46, 57
163		92, 129, 134, 353
165			32, 34, 36
167		17, 312, 404
169		19, 63, 219, 265
172		70, 103, 140, 415
174		40, 42, 165, 289
177		121, 154, 405
179		280
182		45, 47, 66, 218
184		56, 164, 354, 394
185		12, 14, 18, 181, 230, 261, 262, 272, 326, 358, 400
187		34, 44, 46, 49, 51, 72, 130, 264, 297, 316, 427
189			23, 49, 72
191		97, 170, 191, 242, 245, 257, 389
193		81, 115, 135, 274
195		15, 20, 24, 32, 221, 236, 306, 346
197		75, 76, 79, 87, 89, 145, 282
200		82, 88, 177, 237
202		199, 325, 420
204		133, 183, 332, 391
206		13, 69, 290, 381
208		37, 341, 412
210		16, 38, 117, 286
212				2, 3, 4
215		84, 101, 203, 271
216		98, 267, 275, 322
218			17, 20
219			22, 30, 42, 52, 69
222		59, 205, 210, 255, 429
224		104, 385, 414, 423, 424
225			45, 58, 67
227		33, 35, 152, 197, 268, 356, 368
230		55, 155, 194, 291
232		57, 224, 379
233		54, 169, 198, 228
235		162, 240, 285, 298, 305, 376, 387
237		184, 190, 195, 241
239		68, 74, 269, 374
241		25, 31, 287, 302
244		58
246		62, 168, 172, 209, 226, 318, 364, 398
248		167, 359
249		107, 139, 179, 383
251		90, 156, 213, 357

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/ revised November, 2001