Prime p₀= 3565551684219451349 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 10 2 76 2 58 2 70 2 28 2 118 2 (?)
a_n: 0 2 12 14 90 92 150 152 222 224 252 254 372 374
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, 10 76 58 70 28 118

STPT(10)	3565550239289563541	0 2 30 32 78 80 126 128 156 158	-1444929887808
SPT(16)	3565551000643357811	0 12 26 116 150 158 210 236 282 308 360 368 402 492 506 518	-683576093538
SPT(16)	3565551171268725293	0 6 26 54 68 116 128 156 158 186 198 246 260 288 308 314	-512950726056
STPT(10)	3565551370055329397	0 2 30 32 42 44 54 56 84 86	-314164121952
SPT(16)	3565551599083821947	0 12 42 62 66 80 90 174 182 266 276 290 294 314 344 356	-85135629402
TPT(14)	3565551658266092951	0 2 30 32 126 128 150 152 210 212 276 278 288 290	-25953358398
SPT(16)	3565552429821026069	0 24 50 68 78 84 90 122 150 182 188 194 204 222 248 272	745601574720
SPT(16)	3565553280985297517	0 2 12 14 102 104 144 146 180 182 222 224 312 314 324 326	1596765846168
TPT(16)	3565553280985297517	0 2 12 14 102 104 144 146 180 182 222 224 312 314 324 326	1596765846168
STPT(16)	3565553280985297517	0 2 12 14 102 104 144 146 180 182 222 224 312 314 324 326	1596765846168
STPT(10)	3565553555369203751	0 2 78 80 108 110 138 140 216 218	1871149752402