Prime p₀= 3565549567947253979 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 118 2 16 2 40 2 16 2 22 2 16 2 (?)
a_n: 0 2 120 122 138 140 180 182 198 200 222 224 240 242
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: 242
Found by: [SG]KidDoesCrunch
db: batch=84, k=14, 118 16 40 16 22 16

SPT(16)	3565547147492474273	0 6 24 26 38 50 98 120 194 216 264 276 288 290 308 314	-2420454779706
SPT(13)	3565547549360648767	0 12 42 174 180 210 222 234 264 270 402 432 444	-2018586605212
STPT(10)	3565547821167460157	0 2 30 32 42 44 54 56 84 86	-1746779793822
STPT(10)	3565547888519129801	0 2 36 38 78 80 120 122 156 158	-1679428124178
SPT(16)	3565547998326994367	0 30 36 44 114 150 156 210 296 350 356 392 462 470 476 506	-1569620259612
TPT(14)	3565548236343050777	0 2 210 212 240 242 264 266 282 284 330 332 372 374	-1331604203202
SPT(16)	3565550084441904719	0 14 24 72 128 132 150 200 222 272 290 294 350 398 408 422	516494650740
STPT(10)	3565550239289563541	0 2 30 32 78 80 126 128 156 158	671342309562
SPT(16)	3565551000643357811	0 12 26 116 150 158 210 236 282 308 360 368 402 492 506 518	1432696103832
SPT(16)	3565551171268725293	0 6 26 54 68 116 128 156 158 186 198 246 260 288 308 314	1603321471314
STPT(10)	3565551370055329397	0 2 30 32 42 44 54 56 84 86	1802108075418