Prime p₀= 3565551658266092951 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 28 2 94 2 22 2 58 2 64 2 10 2 (?)
a_n: 0 2 30 32 126 128 150 152 210 212 276 278 288 290
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: 290
Found by: [SG]KidDoesCrunch
db: batch=84, k=14, 28 94 22 58 64 10

SPT(16)	3565550084441904719	0 14 24 72 128 132 150 200 222 272 290 294 350 398 408 422	-1573824188232
STPT(10)	3565550239289563541	0 2 30 32 78 80 126 128 156 158	-1418976529410
SPT(16)	3565551000643357811	0 12 26 116 150 158 210 236 282 308 360 368 402 492 506 518	-657622735140
SPT(16)	3565551171268725293	0 6 26 54 68 116 128 156 158 186 198 246 260 288 308 314	-486997367658
STPT(10)	3565551370055329397	0 2 30 32 42 44 54 56 84 86	-288210763554
SPT(16)	3565551599083821947	0 12 42 62 66 80 90 174 182 266 276 290 294 314 344 356	-59182271004
TPT(14)	3565551684219451349	0 2 12 14 90 92 150 152 222 224 252 254 372 374	25953358398
SPT(16)	3565552429821026069	0 24 50 68 78 84 90 122 150 182 188 194 204 222 248 272	771554933118
TPT(16)	3565553280985297517	0 2 12 14 102 104 144 146 180 182 222 224 312 314 324 326	1622719204566
STPT(16)	3565553280985297517	0 2 12 14 102 104 144 146 180 182 222 224 312 314 324 326	1622719204566
SPT(16)	3565553280985297517	0 2 12 14 102 104 144 146 180 182 222 224 312 314 324 326	1622719204566