Prime p₀= 4387558418122750859 starts a Symmetric tuple of 12 consecutive primes, which form 6 twin primes (STPT).
d_n: 0 2 40 2 88 2 64 2 88 2 40 2 (?)
a_n: 0 2 42 44 132 134 198 200 288 290 330 332
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: 332
Found by: [SG]KidDoesCrunch
db: batch=85, k=12, 2 40 2 88 2 64

SPT(16)	4387557479029593563	0 6 14 78 80 104 116 164 270 318 330 354 356 420 428 434	-939093157296
STPT(10)	4387557493581660449	0 2 18 20 30 32 42 44 60 62	-924541090410
SPT(16)	4387557856691139671	0 66 80 90 92 98 132 168 290 326 360 366 368 378 392 458	-561431611188
SPT(16)	4387558040857379477	0 6 20 116 146 192 194 200 216 222 224 270 300 396 410 416	-377265371382
STPT(10)	4387558173840327527	0 2 30 32 42 44 54 56 84 86	-244282423332
TPT(14)	4387558270793639141	0 2 30 32 90 92 156 158 198 200 210 212 228 230	-147329111718
STPT(10)	4387558689112458161	0 2 30 32 78 80 126 128 156 158	270989707302
SPT(16)	4387558714903101971	0 32 80 116 128 182 266 270 308 312 396 450 462 498 546 578	296780351112
SPT(16)	4387558734141245419	0 30 40 60 64 78 120 130 138 148 190 204 208 228 238 268	316018494560
STPT(10)	4387558874969517617	0 2 30 32 42 44 54 56 84 86	456846766758
STPT(10)	4387559061574272767	0 2 30 32 42 44 54 56 84 86	643451521908