Prime p₀= 567540552520603721 starts a Tuple of 12 consecutive primes, which form 6 twin primes (TPT).
d_n: 0 2 76 2 10 2 28 2 46 2 10 2 (?)
a_n: 0 2 78 80 90 92 120 122 168 170 180 182
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: 182
Found by: gemini8
db: batch=71, k=12, 76 10 28 46 10

SPT(14)	567540448662410527	0 24 36 42 76 82 156 160 234 240 274 280 292 316	-103858193194
SPT(14)	567540453297704417	0 6 14 24 66 110 144 146 180 224 266 276 284 290	-99222899304
SPT(14)	567540458328241729	0 42 48 88 100 118 154 288 324 342 354 394 400 442	-94192361992
SPT(14)	567540519847167827	0 30 36 50 80 200 204 206 210 330 360 374 380 410	-32673435894
SPT(14)	567540526077169157	0 2 14 42 44 50 102 104 156 162 164 192 204 206	-26443434564
SPT(14)	567540526975788721	0 6 40 60 88 90 106 192 208 210 238 258 292 298	-25544815000
SPT(14)	567540552859746541	0 12 28 60 108 132 136 162 166 190 238 270 286 298	339142820
SPT(14)	567540578699045311	0 6 28 52 66 90 96 112 118 142 156 180 202 208	26178441590
SPT(14)	567540583257557303	0 8 14 18 38 60 78 80 98 120 140 144 150 158	30736953582
SPT(14)	567540584001839779	0 22 24 64 84 88 114 148 174 178 198 238 240 262	31481236058
TPT(12)	567540594944513171	0 2 78 80 120 122 126 128 156 158 198 200	42423909450