Every node has six lines. Every line has six nodes. If you like these designs You can try out my codes.
base={{0,-1},{0,Root[1+2 #1+2 #1^3+#1^4&,2]},{1/2,Root[-11-8 #1+24 #1^2-32 #1^3+16 #1^4&,1]},{1/2,Root[-11+8 #1+24 #1^2+32 #1^3+16 #1^4&,2]},{1/2 (2-Sqrt[3]),-(1/2)},{Root[-1+4 #1-8 #1^3+8 #1^4&,2],Root[3-24 #1+48 #1^2-24 #1^3+8 #1^4&,1]},{Root[-1+4 #1-8 #1^3+8 #1^4&,2],Root[3+24 #1+48 #1^2+24 #1^3+8 #1^4&,2]},{Root[1+4 #1+16 #1^3+16 #1^4&,2],Root[1-20 #1+48 #1^2+16 #1^3+16 #1^4&,1]}};
nodes = Flatten[Table[RootReduce[#.RotationMatrix[n Pi/6 ]], {n, 0, 11}] & /@ base,1];
lines={{1,9,60,59,26,38},{1,8,49,57,83,64},{1,6,54,50,76,69},{1,5,52,51,30,42},{1,24,37,41,84,73},{1,14,27,31,67,68},{12,8,59,58,25,37},{12,7,60,56,82,63},{12,5,49,53,75,68},{12,4,51,50,29,41},{12,13,26,30,66,67},{12,23,48,40,83,84},{11,7,58,57,36,48},{11,6,59,55,81,62},{11,4,60,52,74,67},{11,3,49,50,28,40},{11,24,25,29,66,65},{11,22,47,39,82,83},{10,6,57,56,35,47},{10,5,58,54,80,61},{10,3,59,51,73,66},{10,2,49,60,27,39},{10,23,36,28,64,65},{10,21,46,38,81,82},{9,5,56,55,34,46},{9,4,57,53,79,72},{9,2,58,50,84,65},{9,22,35,27,63,64},{9,20,45,37,80,81},{8,4,55,54,33,45},{8,3,56,52,78,71},{8,21,34,26,62,63},{8,19,44,48,80,79},{7,3,54,53,32,44},{7,2,55,51,77,70},{7,20,33,25,61,62},{7,18,43,47,78,79},{6,2,53,52,31,43},{6,19,32,36,72,61},{6,17,42,46,77,78},{5,18,35,31,71,72},{5,16,45,41,76,77},{4,17,34,30,70,71},{4,15,44,40,75,76},{3,16,33,29,69,70},{3,14,43,39,74,75},{2,13,42,38,73,74},{2,15,32,28,68,69},{49,92,48,41,82,74},{49,94,26,31,69,65},{60,93,25,30,68,64},{60,91,47,40,81,73},{59,92,36,29,67,63},{59,90,46,39,80,84},{58,91,35,28,66,62},{58,89,45,38,83,79},{57,90,34,27,61,65},{57,88,44,37,82,78},{56,89,33,26,72,64},{56,87,43,48,81,77},{55,88,32,25,71,63},{55,86,42,47,80,76},{54,87,36,31,70,62},{54,85,46,41,75,79},{53,86,35,30,69,61},{53,96,45,40,74,78},{52,85,34,29,68,72},{52,95,44,39,73,77},{51,96,33,28,67,71},{51,94,43,38,84,76},{50,93,42,37,83,75},{50,95,32,27,66,70},{13,21,92,91,27,37},{13,20,93,89,82,65},{13,18,86,94,75,70},{13,17,96,95,31,41},{24,20,91,90,26,48},{24,19,92,88,81,64},{24,17,93,85,74,69},{24,16,95,94,30,40},{23,19,90,89,25,47},{23,18,91,87,80,63},{23,16,92,96,73,68},{23,15,93,94,29,39},{22,18,89,88,36,46},{22,17,90,86,79,62},{22,15,91,95,84,67},{22,14,93,92,28,38},{21,17,88,87,35,45},{21,16,89,85,78,61},{21,14,90,94,83,66},{20,16,87,86,34,44},{20,15,88,96,77,72},{19,15,86,85,33,43},{19,14,87,95,76,71},{18,14,85,96,32,42}};
From L. W. Berman, "Geometric Constructions for Symmetric 6-Configurations," Rigidity and Symmetry, Springer, 2014, p. 83.
