Prime p₀= 4506178279877500199 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 28 2 40 2 4 2 70 2 100 2 34 2 (?)
a_n: 0 2 30 32 72 74 78 80 150 152 252 254 288 290
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=85, k=14, 28 40 4 70 100 34

SPT(16)	4506177100403864447	0 6 92 104 144 150 170 204 212 246 266 272 312 324 410 416	-1179473635752
SPT(16)	4506177455676682129	0 4 18 24 54 58 70 84 238 252 264 268 298 304 318 322	-824200818070
STPT(10)	4506177555146461487	0 2 42 44 72 74 102 104 144 146	-724731038712
STPT(10)	4506177643301724359	0 2 12 14 30 32 48 50 60 62	-636575775840
SPT(16)	4506177794270192627	0 12 42 66 74 116 144 164 252 272 300 342 350 374 404 416	-485607307572
STPT(10)	4506178046372409917	0 2 42 44 72 74 102 104 144 146	-233505090282
STPT(10)	4506178308044722277	0 2 90 92 150 152 210 212 300 302	28167222078
SPT(16)	4506178828444766027	0 14 42 44 60 62 86 92 114 120 144 146 162 164 192 206	548567265828
SPT(13)	4506179043011152307	0 36 66 120 150 180 306 432 462 492 546 576 612	763133652108
SPT(16)	4506180325787241371	0 26 50 62 72 86 90 132 140 182 186 200 210 222 246 272	2045909741172
SPT(16)	4506180773346401771	0 48 62 72 108 138 150 152 186 188 200 230 266 276 290 338	2493468901572