direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C7⋊C3, C21⋊3C12, C42.5C6, (C7×Dic3)⋊C3, C7⋊2(C3×Dic3), C14.2(C3×S3), C3⋊(C4×C7⋊C3), C6.(C2×C7⋊C3), C2.(S3×C7⋊C3), (C3×C7⋊C3)⋊3C4, (C6×C7⋊C3).3C2, (C2×C7⋊C3).2S3, SmallGroup(252,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — Dic3×C7⋊C3 |
Generators and relations for Dic3×C7⋊C3
G = < a,b,c,d | a6=c7=d3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of Dic3×C7⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 7A | 7B | 12A | 12B | 12C | 12D | 14A | 14B | 21A | 21B | 28A | 28B | 28C | 28D | 42A | 42B | |
| size | 1 | 1 | 2 | 7 | 7 | 14 | 14 | 3 | 3 | 2 | 7 | 7 | 14 | 14 | 3 | 3 | 21 | 21 | 21 | 21 | 3 | 3 | 6 | 6 | 9 | 9 | 9 | 9 | 6 | 6 | |
| 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 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | i | -i | -1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 12 |
| ρ10 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | i | -i | -1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 12 |
| ρ11 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -i | i | -1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | linear of order 12 |
| ρ12 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -i | i | -1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | linear of order 12 |
| ρ13 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | complex lifted from C3×S3 |
| ρ16 | 2 | -2 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 1 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3×Dic3 |
| ρ17 | 2 | 2 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | complex lifted from C3×S3 |
| ρ18 | 2 | -2 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 1 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3×Dic3 |
| ρ19 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ20 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ21 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ22 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ23 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | -3i | 3i | -3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | -1-√-7/2 | -1+√-7/2 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | 1+√-7/2 | 1-√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ24 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | -3i | 3i | -3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | -1+√-7/2 | -1-√-7/2 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | 1-√-7/2 | 1+√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ25 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 3i | -3i | -3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | -1+√-7/2 | -1-√-7/2 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | 1-√-7/2 | 1+√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ26 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 3i | -3i | -3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | -1-√-7/2 | -1+√-7/2 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | 1+√-7/2 | 1-√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ27 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | 1-√-7 | 1+√-7 | 1-√-7/2 | 1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ28 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | 1+√-7 | 1-√-7 | 1+√-7/2 | 1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ29 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 1-√-7/2 | 1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ30 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 1+√-7/2 | 1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | complex lifted from S3×C7⋊C3 |
►On 84 points
Generators in S84
(1 36 8 22 15 29)(2 37 9 23 16 30)(3 38 10 24 17 31)(4 39 11 25 18 32)(5 40 12 26 19 33)(6 41 13 27 20 34)(7 42 14 28 21 35)(43 71 57 64 50 78)(44 72 58 65 51 79)(45 73 59 66 52 80)(46 74 60 67 53 81)(47 75 61 68 54 82)(48 76 62 69 55 83)(49 77 63 70 56 84)
(1 64 22 43)(2 65 23 44)(3 66 24 45)(4 67 25 46)(5 68 26 47)(6 69 27 48)(7 70 28 49)(8 71 29 50)(9 72 30 51)(10 73 31 52)(11 74 32 53)(12 75 33 54)(13 76 34 55)(14 77 35 56)(15 78 36 57)(16 79 37 58)(17 80 38 59)(18 81 39 60)(19 82 40 61)(20 83 41 62)(21 84 42 63)
(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)
(1 8 15)(2 10 19)(3 12 16)(4 14 20)(5 9 17)(6 11 21)(7 13 18)(22 29 36)(23 31 40)(24 33 37)(25 35 41)(26 30 38)(27 32 42)(28 34 39)(43 50 57)(44 52 61)(45 54 58)(46 56 62)(47 51 59)(48 53 63)(49 55 60)(64 71 78)(65 73 82)(66 75 79)(67 77 83)(68 72 80)(69 74 84)(70 76 81)
G:=sub<Sym(84)| (1,36,8,22,15,29)(2,37,9,23,16,30)(3,38,10,24,17,31)(4,39,11,25,18,32)(5,40,12,26,19,33)(6,41,13,27,20,34)(7,42,14,28,21,35)(43,71,57,64,50,78)(44,72,58,65,51,79)(45,73,59,66,52,80)(46,74,60,67,53,81)(47,75,61,68,54,82)(48,76,62,69,55,83)(49,77,63,70,56,84), (1,64,22,43)(2,65,23,44)(3,66,24,45)(4,67,25,46)(5,68,26,47)(6,69,27,48)(7,70,28,49)(8,71,29,50)(9,72,30,51)(10,73,31,52)(11,74,32,53)(12,75,33,54)(13,76,34,55)(14,77,35,56)(15,78,36,57)(16,79,37,58)(17,80,38,59)(18,81,39,60)(19,82,40,61)(20,83,41,62)(21,84,42,63), (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), (1,8,15)(2,10,19)(3,12,16)(4,14,20)(5,9,17)(6,11,21)(7,13,18)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,50,57)(44,52,61)(45,54,58)(46,56,62)(47,51,59)(48,53,63)(49,55,60)(64,71,78)(65,73,82)(66,75,79)(67,77,83)(68,72,80)(69,74,84)(70,76,81)>;
G:=Group( (1,36,8,22,15,29)(2,37,9,23,16,30)(3,38,10,24,17,31)(4,39,11,25,18,32)(5,40,12,26,19,33)(6,41,13,27,20,34)(7,42,14,28,21,35)(43,71,57,64,50,78)(44,72,58,65,51,79)(45,73,59,66,52,80)(46,74,60,67,53,81)(47,75,61,68,54,82)(48,76,62,69,55,83)(49,77,63,70,56,84), (1,64,22,43)(2,65,23,44)(3,66,24,45)(4,67,25,46)(5,68,26,47)(6,69,27,48)(7,70,28,49)(8,71,29,50)(9,72,30,51)(10,73,31,52)(11,74,32,53)(12,75,33,54)(13,76,34,55)(14,77,35,56)(15,78,36,57)(16,79,37,58)(17,80,38,59)(18,81,39,60)(19,82,40,61)(20,83,41,62)(21,84,42,63), (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), (1,8,15)(2,10,19)(3,12,16)(4,14,20)(5,9,17)(6,11,21)(7,13,18)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,50,57)(44,52,61)(45,54,58)(46,56,62)(47,51,59)(48,53,63)(49,55,60)(64,71,78)(65,73,82)(66,75,79)(67,77,83)(68,72,80)(69,74,84)(70,76,81) );
G=PermutationGroup([[(1,36,8,22,15,29),(2,37,9,23,16,30),(3,38,10,24,17,31),(4,39,11,25,18,32),(5,40,12,26,19,33),(6,41,13,27,20,34),(7,42,14,28,21,35),(43,71,57,64,50,78),(44,72,58,65,51,79),(45,73,59,66,52,80),(46,74,60,67,53,81),(47,75,61,68,54,82),(48,76,62,69,55,83),(49,77,63,70,56,84)], [(1,64,22,43),(2,65,23,44),(3,66,24,45),(4,67,25,46),(5,68,26,47),(6,69,27,48),(7,70,28,49),(8,71,29,50),(9,72,30,51),(10,73,31,52),(11,74,32,53),(12,75,33,54),(13,76,34,55),(14,77,35,56),(15,78,36,57),(16,79,37,58),(17,80,38,59),(18,81,39,60),(19,82,40,61),(20,83,41,62),(21,84,42,63)], [(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)], [(1,8,15),(2,10,19),(3,12,16),(4,14,20),(5,9,17),(6,11,21),(7,13,18),(22,29,36),(23,31,40),(24,33,37),(25,35,41),(26,30,38),(27,32,42),(28,34,39),(43,50,57),(44,52,61),(45,54,58),(46,56,62),(47,51,59),(48,53,63),(49,55,60),(64,71,78),(65,73,82),(66,75,79),(67,77,83),(68,72,80),(69,74,84),(70,76,81)]])
Matrix representation of Dic3×C7⋊C3 ►in GL5(𝔽337)
| 129 | 0 | 0 | 0 | 0 |
| 12 | 209 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 249 | 312 | 0 | 0 | 0 |
| 175 | 88 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 336 | 124 | 1 |
| 0 | 0 | 0 | 124 | 1 |
| 0 | 0 | 336 | 125 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 125 | 1 | 213 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 212 |
G:=sub<GL(5,GF(337))| [129,12,0,0,0,0,209,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[249,175,0,0,0,312,88,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,124,124,125,0,0,1,1,1],[1,0,0,0,0,0,1,0,0,0,0,0,125,1,1,0,0,1,0,1,0,0,213,0,212] >;
Dic3×C7⋊C3 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_7\rtimes C_3 % in TeX
G:=Group("Dic3xC7:C3"); // GroupNames label
G:=SmallGroup(252,17);
// by ID
G=gap.SmallGroup(252,17);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,909]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^7=d^3=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export
Subgroup lattice of Dic3×C7⋊C3 in TeX
Character table of Dic3×C7⋊C3 in TeX