Prime p₀= 408432723827154149 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 88 2 40 2 34 2 10 2 28 2 10 2 (?)
a_n: 0 2 90 92 132 134 168 170 180 182 210 212 222 224
All 14 integers p_n = p₀+a_n = p_n-1+d_n are prime and there are no other primes in the gaps.
diameter: 224
Found by: Cruncher Pete
db: batch=70, k=14, 88 40 34 10 28 10

SPT(14)	408432680228231051	0 2 8 72 78 92 180 200 288 302 308 372 378 380	-43598923098
SPT(14)	408432680917286261	0 12 38 42 66 78 92 96 110 122 146 150 176 188	-42909867888
SPT(14)	408432703694726143	0 10 28 34 58 70 130 138 198 210 234 240 258 268	-20132428006
SPT(14)	408432707433299287	0 22 60 72 82 84 102 124 142 144 154 166 204 226	-16393854862
TPT(12)	408432708080224757	0 2 24 26 72 74 114 116 180 182 210 212	-15746929392
SPT(14)	408432708958520663	0 60 84 104 186 198 230 264 296 308 390 410 434 494	-14868633486
SPT(14)	408432754891437907	0 12 24 42 52 54 70 96 112 114 124 142 154 166	31064283758
SPT(16)	408432763463038339	0 12 18 54 78 100 198 202 210 214 312 334 358 394 400 412	39635884190
TPT(12)	408432780767004611	0 2 48 50 60 62 90 92 126 128 198 200	56939850462
SPT(16)	408432819428012719	0 42 70 72 88 112 162 172 228 238 288 312 328 330 358 400	95600858570
SPT(14)	408432823280455613	0 14 24 38 78 84 90 248 254 260 300 314 324 338	99453301464