Prime p₀= 3565548236343050777 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 208 2 28 2 22 2 16 2 46 2 40 2 (?)
a_n: 0 2 210 212 240 242 264 266 282 284 330 332 372 374
All 14 integers p_n = p₀+a_n = p_n-1+d_n are prime and there are no other primes in the gaps.
diameter: 374
Found by: [SG]KidDoesCrunch
db: batch=84, k=14, 208 28 22 16 46 40

SPT(16)	3565547134303416277	0 10 36 42 72 100 114 126 130 142 156 184 214 220 246 256	-1102039634500
SPT(16)	3565547147492474273	0 6 24 26 38 50 98 120 194 216 264 276 288 290 308 314	-1088850576504
SPT(13)	3565547549360648767	0 12 42 174 180 210 222 234 264 270 402 432 444	-686982402010
STPT(10)	3565547821167460157	0 2 30 32 42 44 54 56 84 86	-415175590620
STPT(10)	3565547888519129801	0 2 36 38 78 80 120 122 156 158	-347823920976
SPT(16)	3565547998326994367	0 30 36 44 114 150 156 210 296 350 356 392 462 470 476 506	-238016056410
TPT(14)	3565549567947253979	0 2 120 122 138 140 180 182 198 200 222 224 240 242	1331604203202
SPT(16)	3565550084441904719	0 14 24 72 128 132 150 200 222 272 290 294 350 398 408 422	1848098853942
STPT(10)	3565550239289563541	0 2 30 32 78 80 126 128 156 158	2002946512764
SPT(16)	3565551000643357811	0 12 26 116 150 158 210 236 282 308 360 368 402 492 506 518	2764300307034
SPT(16)	3565551171268725293	0 6 26 54 68 116 128 156 158 186 198 246 260 288 308 314	2934925674516