metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.80D6, C4⋊Q8⋊4S3, C3⋊C8.8D4, C4.18(S3×D4), C12.38(C2×D4), C3⋊3(C8.2D4), (C2×Q8).70D6, (C2×C12).295D4, C6.24(C4⋊1D4), C42⋊7S3.8C2, (C6×Q8).64C22, C2.15(C12⋊3D4), (C2×C12).404C23, C42.S3⋊14C2, (C4×C12).133C22, C6.96(C8.C22), (C2×D12).108C22, C2.17(Q8.11D6), (C2×Dic6).113C22, (C3×C4⋊Q8)⋊4C2, (C2×C3⋊Q16)⋊15C2, (C2×C6).535(C2×D4), (C2×C4).73(C3⋊D4), (C2×C3⋊C8).137C22, (C2×Q8⋊2S3).7C2, (C2×C4).501(C22×S3), C22.207(C2×C3⋊D4), SmallGroup(192,645)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.80D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >
Subgroups: 368 in 124 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C2×C3⋊C8, D6⋊C4, Q8⋊2S3, C3⋊Q16, C4×C12, C3×C4⋊C4, C2×Dic6, C2×D12, C6×Q8, C8.2D4, C42.S3, C42⋊7S3, C2×Q8⋊2S3, C2×C3⋊Q16, C3×C4⋊Q8, C42.80D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C8.C22, S3×D4, C2×C3⋊D4, C8.2D4, C12⋊3D4, Q8.11D6, C42.80D6
Character table of C42.80D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | complex lifted from Q8.11D6 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from Q8.11D6 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from Q8.11D6 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | complex lifted from Q8.11D6 |
►On 96 points
Generators in S96
(1 68 75 27)(2 28 76 69)(3 70 77 29)(4 30 78 71)(5 72 79 31)(6 32 80 61)(7 62 81 33)(8 34 82 63)(9 64 83 35)(10 36 84 65)(11 66 73 25)(12 26 74 67)(13 45 86 58)(14 59 87 46)(15 47 88 60)(16 49 89 48)(17 37 90 50)(18 51 91 38)(19 39 92 52)(20 53 93 40)(21 41 94 54)(22 55 95 42)(23 43 96 56)(24 57 85 44)
(1 14 7 20)(2 21 8 15)(3 16 9 22)(4 23 10 17)(5 18 11 24)(6 13 12 19)(25 44 31 38)(26 39 32 45)(27 46 33 40)(28 41 34 47)(29 48 35 42)(30 43 36 37)(49 64 55 70)(50 71 56 65)(51 66 57 72)(52 61 58 67)(53 68 59 62)(54 63 60 69)(73 85 79 91)(74 92 80 86)(75 87 81 93)(76 94 82 88)(77 89 83 95)(78 96 84 90)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 6 14 13 7 12 20 19)(2 18 21 11 8 24 15 5)(3 4 16 23 9 10 22 17)(25 69 44 54 31 63 38 60)(26 59 39 62 32 53 45 68)(27 67 46 52 33 61 40 58)(28 57 41 72 34 51 47 66)(29 65 48 50 35 71 42 56)(30 55 43 70 36 49 37 64)(73 82 85 88 79 76 91 94)(74 93 92 75 80 87 86 81)(77 78 89 96 83 84 95 90)
G:=sub<Sym(96)| (1,68,75,27)(2,28,76,69)(3,70,77,29)(4,30,78,71)(5,72,79,31)(6,32,80,61)(7,62,81,33)(8,34,82,63)(9,64,83,35)(10,36,84,65)(11,66,73,25)(12,26,74,67)(13,45,86,58)(14,59,87,46)(15,47,88,60)(16,49,89,48)(17,37,90,50)(18,51,91,38)(19,39,92,52)(20,53,93,40)(21,41,94,54)(22,55,95,42)(23,43,96,56)(24,57,85,44), (1,14,7,20)(2,21,8,15)(3,16,9,22)(4,23,10,17)(5,18,11,24)(6,13,12,19)(25,44,31,38)(26,39,32,45)(27,46,33,40)(28,41,34,47)(29,48,35,42)(30,43,36,37)(49,64,55,70)(50,71,56,65)(51,66,57,72)(52,61,58,67)(53,68,59,62)(54,63,60,69)(73,85,79,91)(74,92,80,86)(75,87,81,93)(76,94,82,88)(77,89,83,95)(78,96,84,90), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,14,13,7,12,20,19)(2,18,21,11,8,24,15,5)(3,4,16,23,9,10,22,17)(25,69,44,54,31,63,38,60)(26,59,39,62,32,53,45,68)(27,67,46,52,33,61,40,58)(28,57,41,72,34,51,47,66)(29,65,48,50,35,71,42,56)(30,55,43,70,36,49,37,64)(73,82,85,88,79,76,91,94)(74,93,92,75,80,87,86,81)(77,78,89,96,83,84,95,90)>;
G:=Group( (1,68,75,27)(2,28,76,69)(3,70,77,29)(4,30,78,71)(5,72,79,31)(6,32,80,61)(7,62,81,33)(8,34,82,63)(9,64,83,35)(10,36,84,65)(11,66,73,25)(12,26,74,67)(13,45,86,58)(14,59,87,46)(15,47,88,60)(16,49,89,48)(17,37,90,50)(18,51,91,38)(19,39,92,52)(20,53,93,40)(21,41,94,54)(22,55,95,42)(23,43,96,56)(24,57,85,44), (1,14,7,20)(2,21,8,15)(3,16,9,22)(4,23,10,17)(5,18,11,24)(6,13,12,19)(25,44,31,38)(26,39,32,45)(27,46,33,40)(28,41,34,47)(29,48,35,42)(30,43,36,37)(49,64,55,70)(50,71,56,65)(51,66,57,72)(52,61,58,67)(53,68,59,62)(54,63,60,69)(73,85,79,91)(74,92,80,86)(75,87,81,93)(76,94,82,88)(77,89,83,95)(78,96,84,90), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,14,13,7,12,20,19)(2,18,21,11,8,24,15,5)(3,4,16,23,9,10,22,17)(25,69,44,54,31,63,38,60)(26,59,39,62,32,53,45,68)(27,67,46,52,33,61,40,58)(28,57,41,72,34,51,47,66)(29,65,48,50,35,71,42,56)(30,55,43,70,36,49,37,64)(73,82,85,88,79,76,91,94)(74,93,92,75,80,87,86,81)(77,78,89,96,83,84,95,90) );
G=PermutationGroup([[(1,68,75,27),(2,28,76,69),(3,70,77,29),(4,30,78,71),(5,72,79,31),(6,32,80,61),(7,62,81,33),(8,34,82,63),(9,64,83,35),(10,36,84,65),(11,66,73,25),(12,26,74,67),(13,45,86,58),(14,59,87,46),(15,47,88,60),(16,49,89,48),(17,37,90,50),(18,51,91,38),(19,39,92,52),(20,53,93,40),(21,41,94,54),(22,55,95,42),(23,43,96,56),(24,57,85,44)], [(1,14,7,20),(2,21,8,15),(3,16,9,22),(4,23,10,17),(5,18,11,24),(6,13,12,19),(25,44,31,38),(26,39,32,45),(27,46,33,40),(28,41,34,47),(29,48,35,42),(30,43,36,37),(49,64,55,70),(50,71,56,65),(51,66,57,72),(52,61,58,67),(53,68,59,62),(54,63,60,69),(73,85,79,91),(74,92,80,86),(75,87,81,93),(76,94,82,88),(77,89,83,95),(78,96,84,90)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,6,14,13,7,12,20,19),(2,18,21,11,8,24,15,5),(3,4,16,23,9,10,22,17),(25,69,44,54,31,63,38,60),(26,59,39,62,32,53,45,68),(27,67,46,52,33,61,40,58),(28,57,41,72,34,51,47,66),(29,65,48,50,35,71,42,56),(30,55,43,70,36,49,37,64),(73,82,85,88,79,76,91,94),(74,93,92,75,80,87,86,81),(77,78,89,96,83,84,95,90)]])
Matrix representation of C42.80D6 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 60 | 60 | 47 |
| 0 | 0 | 13 | 43 | 26 | 13 |
| 0 | 0 | 43 | 13 | 43 | 13 |
| 0 | 0 | 60 | 30 | 60 | 30 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 71 | 0 |
| 0 | 0 | 0 | 72 | 0 | 71 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 4 | 47 | 55 |
| 0 | 0 | 69 | 65 | 18 | 65 |
| 0 | 0 | 64 | 60 | 4 | 69 |
| 0 | 0 | 13 | 4 | 4 | 8 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 22 | 18 | 26 |
| 0 | 0 | 0 | 51 | 8 | 55 |
| 0 | 0 | 64 | 60 | 4 | 69 |
| 0 | 0 | 69 | 9 | 65 | 69 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,30,13,43,60,0,0,60,43,13,30,0,0,60,26,43,60,0,0,47,13,13,30],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,1,0,0,71,0,1,0,0,0,0,71,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,69,69,64,13,0,0,4,65,60,4,0,0,47,18,4,4,0,0,55,65,69,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,22,0,64,69,0,0,22,51,60,9,0,0,18,8,4,65,0,0,26,55,69,69] >;
C42.80D6 in GAP, Magma, Sage, TeX
C_4^2._{80}D_6 % in TeX
G:=Group("C4^2.80D6"); // GroupNames label
G:=SmallGroup(192,645);
// by ID
G=gap.SmallGroup(192,645);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations
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