metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C21⋊2D4, C3⋊2D28, Dic3⋊D7, D14⋊2S3, D42⋊3C2, C14.5D6, C6.5D14, C42.5C22, (C6×D7)⋊2C2, C7⋊1(C3⋊D4), C2.5(S3×D7), (C7×Dic3)⋊3C2, SmallGroup(168,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D28
G = < a,b,c | a3=b28=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C3⋊D28
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 28A | 28B | 28C | 28D | 28E | 28F | 42A | 42B | 42C | |
| size | 1 | 1 | 14 | 42 | 2 | 6 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | 0 | -1 | 0 | -1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ4ζ75-ζ4ζ72 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | -ζ4ζ75+ζ4ζ72 | ζ43ζ76-ζ43ζ7 | -ζ43ζ76+ζ43ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D28 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ4ζ74+ζ4ζ73 | ζ43ζ76-ζ43ζ7 | -ζ43ζ76+ζ43ζ7 | ζ4ζ74-ζ4ζ73 | -ζ4ζ75+ζ4ζ72 | ζ4ζ75-ζ4ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D28 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ43ζ76-ζ43ζ7 | ζ4ζ75-ζ4ζ72 | -ζ4ζ75+ζ4ζ72 | -ζ43ζ76+ζ43ζ7 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D28 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ43ζ76+ζ43ζ7 | -ζ4ζ75+ζ4ζ72 | ζ4ζ75-ζ4ζ72 | ζ43ζ76-ζ43ζ7 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D28 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ4ζ74-ζ4ζ73 | -ζ43ζ76+ζ43ζ7 | ζ43ζ76-ζ43ζ7 | -ζ4ζ74+ζ4ζ73 | ζ4ζ75-ζ4ζ72 | -ζ4ζ75+ζ4ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D28 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ4ζ75+ζ4ζ72 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | ζ4ζ75-ζ4ζ72 | -ζ43ζ76+ζ43ζ7 | ζ43ζ76-ζ43ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D28 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√-3 | √-3 | 2 | 2 | 2 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √-3 | -√-3 | 2 | 2 | 2 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal faithful |
| ρ23 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ24 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal faithful |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal faithful |
| ρ27 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
►On 84 points
Generators in S84
(1 52 78)(2 79 53)(3 54 80)(4 81 55)(5 56 82)(6 83 29)(7 30 84)(8 57 31)(9 32 58)(10 59 33)(11 34 60)(12 61 35)(13 36 62)(14 63 37)(15 38 64)(16 65 39)(17 40 66)(18 67 41)(19 42 68)(20 69 43)(21 44 70)(22 71 45)(23 46 72)(24 73 47)(25 48 74)(26 75 49)(27 50 76)(28 77 51)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 84)(53 83)(54 82)(55 81)(56 80)
G:=sub<Sym(84)| (1,52,78)(2,79,53)(3,54,80)(4,81,55)(5,56,82)(6,83,29)(7,30,84)(8,57,31)(9,32,58)(10,59,33)(11,34,60)(12,61,35)(13,36,62)(14,63,37)(15,38,64)(16,65,39)(17,40,66)(18,67,41)(19,42,68)(20,69,43)(21,44,70)(22,71,45)(23,46,72)(24,73,47)(25,48,74)(26,75,49)(27,50,76)(28,77,51), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,84)(53,83)(54,82)(55,81)(56,80)>;
G:=Group( (1,52,78)(2,79,53)(3,54,80)(4,81,55)(5,56,82)(6,83,29)(7,30,84)(8,57,31)(9,32,58)(10,59,33)(11,34,60)(12,61,35)(13,36,62)(14,63,37)(15,38,64)(16,65,39)(17,40,66)(18,67,41)(19,42,68)(20,69,43)(21,44,70)(22,71,45)(23,46,72)(24,73,47)(25,48,74)(26,75,49)(27,50,76)(28,77,51), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,84)(53,83)(54,82)(55,81)(56,80) );
G=PermutationGroup([[(1,52,78),(2,79,53),(3,54,80),(4,81,55),(5,56,82),(6,83,29),(7,30,84),(8,57,31),(9,32,58),(10,59,33),(11,34,60),(12,61,35),(13,36,62),(14,63,37),(15,38,64),(16,65,39),(17,40,66),(18,67,41),(19,42,68),(20,69,43),(21,44,70),(22,71,45),(23,46,72),(24,73,47),(25,48,74),(26,75,49),(27,50,76),(28,77,51)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,60),(49,59),(50,58),(51,57),(52,84),(53,83),(54,82),(55,81),(56,80)]])
C3⋊D28 is a maximal subgroup of
D28⋊S3 D6.D14 D14.D6 S3×D28 Dic7.D6 D7×C3⋊D4 D6⋊D14
C3⋊D28 is a maximal quotient of C3⋊D56 C6.D28 C21⋊SD16 C3⋊Dic28 D14⋊Dic3 D42⋊C4 C14.Dic6
Matrix representation of C3⋊D28 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 336 | 336 |
| 219 | 62 | 0 | 0 |
| 299 | 257 | 0 | 0 |
| 0 | 0 | 278 | 139 |
| 0 | 0 | 198 | 59 |
| 0 | 193 | 0 | 0 |
| 227 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 |
| 0 | 0 | 336 | 0 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,0,336,0,0,1,336],[219,299,0,0,62,257,0,0,0,0,278,198,0,0,139,59],[0,227,0,0,193,0,0,0,0,0,0,336,0,0,336,0] >;
C3⋊D28 in GAP, Magma, Sage, TeX
C_3\rtimes D_{28} % in TeX
G:=Group("C3:D28"); // GroupNames label
G:=SmallGroup(168,16);
// by ID
G=gap.SmallGroup(168,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,61,26,168,3604]);
// Polycyclic
G:=Group<a,b,c|a^3=b^28=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3⋊D28 in TeX
Character table of C3⋊D28 in TeX