metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C21⋊1D4, D6⋊1D7, D14⋊1S3, C6.4D14, C14.4D6, Dic21⋊4C2, C42.4C22, (C6×D7)⋊1C2, C7⋊2(C3⋊D4), C3⋊2(C7⋊D4), C2.4(S3×D7), (S3×C14)⋊1C2, SmallGroup(168,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C21⋊D4
G = < a,b,c | a21=b4=c2=1, bab-1=a-1, cac=a13, cbc=b-1 >
Character table of C21⋊D4
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | 7B | 7C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 21A | 21B | 21C | 42A | 42B | 42C | |
| size | 1 | 1 | 6 | 14 | 2 | 42 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 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 | -2 | -1 | 0 | -1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 2 | -1 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ14 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √-3 | -√-3 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75+ζ72 | ζ75-ζ72 | -ζ76+ζ7 | ζ74-ζ73 | -ζ74+ζ73 | ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | complex lifted from C7⋊D4 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75-ζ72 | -ζ75+ζ72 | ζ76-ζ7 | -ζ74+ζ73 | ζ74-ζ73 | -ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | complex lifted from C7⋊D4 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74+ζ73 | ζ74-ζ73 | ζ75-ζ72 | ζ76-ζ7 | -ζ76+ζ7 | -ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | complex lifted from C7⋊D4 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76+ζ7 | ζ76-ζ7 | ζ74-ζ73 | -ζ75+ζ72 | ζ75-ζ72 | -ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | complex lifted from C7⋊D4 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74-ζ73 | -ζ74+ζ73 | -ζ75+ζ72 | -ζ76+ζ7 | ζ76-ζ7 | ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | complex lifted from C7⋊D4 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√-3 | √-3 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76-ζ7 | -ζ76+ζ7 | -ζ74+ζ73 | ζ75-ζ72 | -ζ75+ζ72 | ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | complex lifted from C7⋊D4 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ23 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ24 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | symplectic faithful, Schur index 2 |
►On 84 points
Generators in S84
(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 46 38 84)(2 45 39 83)(3 44 40 82)(4 43 41 81)(5 63 42 80)(6 62 22 79)(7 61 23 78)(8 60 24 77)(9 59 25 76)(10 58 26 75)(11 57 27 74)(12 56 28 73)(13 55 29 72)(14 54 30 71)(15 53 31 70)(16 52 32 69)(17 51 33 68)(18 50 34 67)(19 49 35 66)(20 48 36 65)(21 47 37 64)
(2 14)(3 6)(4 19)(5 11)(7 16)(9 21)(10 13)(12 18)(17 20)(22 40)(23 32)(25 37)(26 29)(27 42)(28 34)(30 39)(33 36)(35 41)(43 66)(44 79)(45 71)(46 84)(47 76)(48 68)(49 81)(50 73)(51 65)(52 78)(53 70)(54 83)(55 75)(56 67)(57 80)(58 72)(59 64)(60 77)(61 69)(62 82)(63 74)
G:=sub<Sym(84)| (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), (1,46,38,84)(2,45,39,83)(3,44,40,82)(4,43,41,81)(5,63,42,80)(6,62,22,79)(7,61,23,78)(8,60,24,77)(9,59,25,76)(10,58,26,75)(11,57,27,74)(12,56,28,73)(13,55,29,72)(14,54,30,71)(15,53,31,70)(16,52,32,69)(17,51,33,68)(18,50,34,67)(19,49,35,66)(20,48,36,65)(21,47,37,64), (2,14)(3,6)(4,19)(5,11)(7,16)(9,21)(10,13)(12,18)(17,20)(22,40)(23,32)(25,37)(26,29)(27,42)(28,34)(30,39)(33,36)(35,41)(43,66)(44,79)(45,71)(46,84)(47,76)(48,68)(49,81)(50,73)(51,65)(52,78)(53,70)(54,83)(55,75)(56,67)(57,80)(58,72)(59,64)(60,77)(61,69)(62,82)(63,74)>;
G:=Group( (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), (1,46,38,84)(2,45,39,83)(3,44,40,82)(4,43,41,81)(5,63,42,80)(6,62,22,79)(7,61,23,78)(8,60,24,77)(9,59,25,76)(10,58,26,75)(11,57,27,74)(12,56,28,73)(13,55,29,72)(14,54,30,71)(15,53,31,70)(16,52,32,69)(17,51,33,68)(18,50,34,67)(19,49,35,66)(20,48,36,65)(21,47,37,64), (2,14)(3,6)(4,19)(5,11)(7,16)(9,21)(10,13)(12,18)(17,20)(22,40)(23,32)(25,37)(26,29)(27,42)(28,34)(30,39)(33,36)(35,41)(43,66)(44,79)(45,71)(46,84)(47,76)(48,68)(49,81)(50,73)(51,65)(52,78)(53,70)(54,83)(55,75)(56,67)(57,80)(58,72)(59,64)(60,77)(61,69)(62,82)(63,74) );
G=PermutationGroup([[(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)], [(1,46,38,84),(2,45,39,83),(3,44,40,82),(4,43,41,81),(5,63,42,80),(6,62,22,79),(7,61,23,78),(8,60,24,77),(9,59,25,76),(10,58,26,75),(11,57,27,74),(12,56,28,73),(13,55,29,72),(14,54,30,71),(15,53,31,70),(16,52,32,69),(17,51,33,68),(18,50,34,67),(19,49,35,66),(20,48,36,65),(21,47,37,64)], [(2,14),(3,6),(4,19),(5,11),(7,16),(9,21),(10,13),(12,18),(17,20),(22,40),(23,32),(25,37),(26,29),(27,42),(28,34),(30,39),(33,36),(35,41),(43,66),(44,79),(45,71),(46,84),(47,76),(48,68),(49,81),(50,73),(51,65),(52,78),(53,70),(54,83),(55,75),(56,67),(57,80),(58,72),(59,64),(60,77),(61,69),(62,82),(63,74)]])
C21⋊D4 is a maximal subgroup of
D28⋊5S3 D6.D14 D12⋊5D7 C28⋊D6 C42.C23 D7×C3⋊D4 S3×C7⋊D4
C21⋊D4 is a maximal quotient of C21⋊D8 C28.D6 C42.D4 C21⋊Q16 D14⋊Dic3 D6⋊Dic7 Dic21⋊C4
Matrix representation of C21⋊D4 ►in GL4(𝔽337) generated by
| 0 | 194 | 0 | 0 |
| 33 | 227 | 0 | 0 |
| 0 | 0 | 208 | 0 |
| 0 | 0 | 84 | 128 |
| 0 | 194 | 0 | 0 |
| 304 | 0 | 0 | 0 |
| 0 | 0 | 206 | 205 |
| 0 | 0 | 33 | 131 |
| 0 | 194 | 0 | 0 |
| 304 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 238 | 336 |
G:=sub<GL(4,GF(337))| [0,33,0,0,194,227,0,0,0,0,208,84,0,0,0,128],[0,304,0,0,194,0,0,0,0,0,206,33,0,0,205,131],[0,304,0,0,194,0,0,0,0,0,1,238,0,0,0,336] >;
C21⋊D4 in GAP, Magma, Sage, TeX
C_{21}\rtimes D_4 % in TeX
G:=Group("C21:D4"); // GroupNames label
G:=SmallGroup(168,15);
// by ID
G=gap.SmallGroup(168,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,61,168,3604]);
// Polycyclic
G:=Group<a,b,c|a^21=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C21⋊D4 in TeX
Character table of C21⋊D4 in TeX