Prime p₀= 4404370615099780067 starts a Symmetric tuple of 12 consecutive primes, which form 6 twin primes (STPT).
d_n: 0 2 10 2 16 2 22 2 16 2 10 2 (?)
a_n: 0 2 12 14 30 32 54 56 72 74 84 86
All 12 integers p_n = p₀+a_n = p_n-1+d_n are prime and there are no other primes in the gaps.
diameter: 86
Found by: [SG]KidDoesCrunch
db: batch=85, k=12, 2 10 2 16 2 22

STPT(10)	4404368636591039717	0 2 42 44 72 74 102 104 144 146	-1978508740350
SPT(16)	4404369083239171499	0 32 50 60 84 98 110 138 224 252 264 278 302 312 330 362	-1531860608568
STPT(12)	4404369556086867581	0 2 30 32 60 62 156 158 186 188 216 218	-1059012912486
SPT(16)	4404369764931021881	0 140 186 198 206 228 240 260 318 338 350 372 380 392 438 578	-850168758186
SPT(16)	4404370101561138547	0 10 46 54 60 72 82 100 126 144 154 166 172 180 216 226	-513538641520
TPT(14)	4404370373255898977	0 2 114 116 150 152 192 194 204 206 252 254 282 284	-241843881090
SPT(16)	4404370787591224819	0 4 40 90 108 130 162 174 208 220 252 274 292 342 378 382	172491444752
STPT(10)	4404370990888880951	0 2 30 32 60 62 90 92 120 122	375789100884
STPT(10)	4404371196072759017	0 2 30 32 42 44 54 56 84 86	580972978950
SPT(16)	4404371295990820157	0 2 12 14 54 56 84 86 150 152 180 182 222 224 234 236	680891040090
TPT(16)	4404371295990820157	0 2 12 14 54 56 84 86 150 152 180 182 222 224 234 236	680891040090