Prime p₀= 378622672263483671 starts a Tuple of 12 consecutive primes, which form 6 twin primes (TPT).
d_n: 0 2 58 2 28 2 148 2 4 2 22 2 (?)
a_n: 0 2 60 62 90 92 240 242 246 248 270 272
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: 272
Found by: Cruncher Pete
db: batch=70, k=12, 58 28 148 4 22

SPT(14)	378622520078323921	0 36 52 58 70 72 130 168 226 228 240 246 262 298	-152185159750
SPT(14)	378622523133399043	0 10 64 78 136 160 168 226 234 258 316 330 384 394	-149130084628
SPT(14)	378622551291800513	0 8 14 20 24 36 38 66 68 80 84 90 96 104	-120971683158
SPT(14)	378622554091685173	0 6 36 64 78 90 106 138 154 166 180 208 238 244	-118171798498
SPT(14)	378622658785374959	0 2 8 30 42 90 102 158 170 218 230 252 258 260	-13478108712
SPT(14)	378622665215605033	0 28 54 90 100 136 138 196 198 234 244 280 306 334	-7047878638
SPT(14)	378622676244095191	0 10 16 30 42 126 142 150 166 250 262 276 282 292	3980611520
SPT(14)	378622682457861329	0 20 42 48 60 78 80 120 122 140 152 158 180 200	10194377658
SPT(14)	378622693254313961	0 18 26 60 78 140 150 176 186 248 266 300 308 326	20990830290
SPT(14)	378622710395435057	0 44 74 180 186 200 270 290 360 374 380 486 516 560	38131951386
SPT(14)	378622721034974107	0 6 30 42 46 70 94 132 156 180 184 196 220 226	48771490436