Prime p₀= 4004327943855599 starts a Tuple of 12 consecutive primes, which form 6 twin primes (TPT).
d_n: 0 2 40 2 46 2 28 2 16 2 22 2 (?)
a_n: 0 2 42 44 90 92 120 122 138 140 162 164
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: 164
Found by: crashtech
db: batch=61, k=12, 40 46 28 16 22

SPT(14)	4004303492228279	0 14 42 48 60 104 108 164 168 212 224 230 258 272	-24451627320
SPT(14)	4004305228732223	0 24 38 44 48 56 60 104 108 116 120 126 140 164	-22715123376
SPT(14)	4004305600918741	0 28 40 42 46 48 60 118 130 132 136 138 150 178	-22342936858
SPT(14)	4004314203689993	0 20 30 44 78 90 110 114 134 146 180 194 204 224	-13740165606
TPT(12)	4004322457574807	0 2 12 14 30 32 54 56 132 134 174 176	-5486280792
TPT(12)	4004326164396587	0 2 30 32 234 236 240 242 252 254 264 266	-1779459012
SPT(14)	4004338180198117	0 34 90 94 100 102 112 132 142 144 150 154 210 244	10236342518
SPT(14)	4004358282678079	0 28 40 54 84 148 154 258 264 328 358 372 384 412	30338822480
SPT(14)	4004358310949189	0 20 30 32 78 92 120 140 168 182 228 230 240 260	30367093590
SPT(14)	4004359408023203	0 26 38 66 74 78 140 144 206 210 218 246 258 284	31464167604
TPT(12)	4004361275412737	0 2 30 32 54 56 114 116 120 122 150 152	33331557138