Prime p₀= 2305774154683905767 starts a Symmetric tuple of 12 consecutive primes, which form 6 twin primes (STPT).
d_n: 0 2 10 2 10 2 4 2 10 2 10 2 (?)
a_n: 0 2 12 14 24 26 30 32 42 44 54 56
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: 56
Found by: Orange Kid
db: batch=81, k=12, 2 10 2 10 2 4

SPT(18)	2305773231392343313	0 6 16 40 54 58 60 84 96 118 130 154 156 160 174 198 208 214	-923291562454
STPT(10)	2305773244476791891	0 2 6 8 48 50 90 92 96 98	-910207113876
SPT(16)	2305773314064481139	0 42 48 54 92 104 110 132 140 162 168 180 218 224 230 272	-840619424628
STPT(10)	2305773456843262907	0 2 54 56 72 74 90 92 144 146	-697840642860
STPT(10)	2305773773166421247	0 2 84 86 132 134 180 182 264 266	-381517484520
TPT(14)	2305774109009540681	0 2 30 32 66 68 108 110 120 122 150 152 210 212	-45674365086
SPT(16)	2305774685848410571	0 58 72 126 132 168 246 270 298 322 400 436 442 496 510 568	531164504804
STPT(10)	2305774764711194597	0 2 60 62 72 74 84 86 144 146	610027288830
SPT(16)	2305775136581385857	0 26 42 114 126 134 152 192 224 264 282 290 302 374 390 416	981897480090
TPT(14)	2305775379100119671	0 2 6 8 90 92 96 98 120 122 138 140 180 182	1224416213904
SPT(16)	2305775529721894127	0 30 42 50 72 122 144 156 170 182 204 254 276 284 296 326	1375037988360