metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C21⋊3D4, C7⋊2D12, D6⋊2D7, Dic7⋊S3, D42⋊4C2, C6.6D14, C14.6D6, C42.6C22, C3⋊1(C7⋊D4), C2.6(S3×D7), (S3×C14)⋊2C2, (C3×Dic7)⋊3C2, SmallGroup(168,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊D12
G = < a,b,c | a7=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C7⋊D12
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 7A | 7B | 7C | 12A | 12B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 21A | 21B | 21C | 42A | 42B | 42C | |
| size | 1 | 1 | 6 | 42 | 2 | 14 | 2 | 2 | 2 | 2 | 14 | 14 | 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 | 0 | -1 | -2 | -1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | 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 | 2 | 2 | 2 | 0 | 0 | -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 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | 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 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ11 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ12 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | 2 | 2 | 2 | √3 | -√3 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D12 |
| ρ13 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ14 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | 2 | 2 | 2 | -√3 | √3 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74-ζ73 | -ζ74+ζ73 | ζ76-ζ7 | -ζ75+ζ72 | ζ75-ζ72 | -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 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75+ζ72 | ζ75-ζ72 | -ζ74+ζ73 | -ζ76+ζ7 | ζ76-ζ7 | ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | complex lifted from C7⋊D4 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76+ζ7 | ζ76-ζ7 | ζ75-ζ72 | ζ74-ζ73 | -ζ74+ζ73 | -ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | complex lifted from C7⋊D4 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75-ζ72 | -ζ75+ζ72 | ζ74-ζ73 | ζ76-ζ7 | -ζ76+ζ7 | -ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | complex lifted from C7⋊D4 |
| ρ20 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74+ζ73 | ζ74-ζ73 | -ζ76+ζ7 | ζ75-ζ72 | -ζ75+ζ72 | ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | complex lifted from C7⋊D4 |
| ρ21 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76-ζ7 | -ζ76+ζ7 | -ζ75+ζ72 | -ζ74+ζ73 | ζ74-ζ73 | ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | complex lifted from C7⋊D4 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 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 |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | -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 faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | -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 faithful, Schur index 2 |
| ρ25 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 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 |
| ρ26 | 4 | 4 | 0 | 0 | -2 | 0 | -2 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 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 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | -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 faithful, Schur index 2 |
►On 84 points
Generators in S84
(1 18 80 54 71 39 33)(2 34 40 72 55 81 19)(3 20 82 56 61 41 35)(4 36 42 62 57 83 21)(5 22 84 58 63 43 25)(6 26 44 64 59 73 23)(7 24 74 60 65 45 27)(8 28 46 66 49 75 13)(9 14 76 50 67 47 29)(10 30 48 68 51 77 15)(11 16 78 52 69 37 31)(12 32 38 70 53 79 17)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)(37 78)(38 77)(39 76)(40 75)(41 74)(42 73)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
G:=sub<Sym(84)| (1,18,80,54,71,39,33)(2,34,40,72,55,81,19)(3,20,82,56,61,41,35)(4,36,42,62,57,83,21)(5,22,84,58,63,43,25)(6,26,44,64,59,73,23)(7,24,74,60,65,45,27)(8,28,46,66,49,75,13)(9,14,76,50,67,47,29)(10,30,48,68,51,77,15)(11,16,78,52,69,37,31)(12,32,38,70,53,79,17), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)>;
G:=Group( (1,18,80,54,71,39,33)(2,34,40,72,55,81,19)(3,20,82,56,61,41,35)(4,36,42,62,57,83,21)(5,22,84,58,63,43,25)(6,26,44,64,59,73,23)(7,24,74,60,65,45,27)(8,28,46,66,49,75,13)(9,14,76,50,67,47,29)(10,30,48,68,51,77,15)(11,16,78,52,69,37,31)(12,32,38,70,53,79,17), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61) );
G=PermutationGroup([[(1,18,80,54,71,39,33),(2,34,40,72,55,81,19),(3,20,82,56,61,41,35),(4,36,42,62,57,83,21),(5,22,84,58,63,43,25),(6,26,44,64,59,73,23),(7,24,74,60,65,45,27),(8,28,46,66,49,75,13),(9,14,76,50,67,47,29),(10,30,48,68,51,77,15),(11,16,78,52,69,37,31),(12,32,38,70,53,79,17)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35),(37,78),(38,77),(39,76),(40,75),(41,74),(42,73),(43,84),(44,83),(45,82),(46,81),(47,80),(48,79),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61)]])
C7⋊D12 is a maximal subgroup of
D12⋊D7 D84⋊C2 D6.D14 D7×D12 Dic3.D14 S3×C7⋊D4 D6⋊D14
C7⋊D12 is a maximal quotient of C7⋊D24 D12.D7 Dic6⋊D7 C7⋊Dic12 D6⋊Dic7 D42⋊C4 C42.Q8
Matrix representation of C7⋊D12 ►in GL4(𝔽337) generated by
| 0 | 1 | 0 | 0 |
| 336 | 303 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 122 | 27 | 0 | 0 |
| 260 | 215 | 0 | 0 |
| 0 | 0 | 292 | 306 |
| 0 | 0 | 87 | 0 |
| 1 | 0 | 0 | 0 |
| 303 | 336 | 0 | 0 |
| 0 | 0 | 336 | 290 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(337))| [0,336,0,0,1,303,0,0,0,0,1,0,0,0,0,1],[122,260,0,0,27,215,0,0,0,0,292,87,0,0,306,0],[1,303,0,0,0,336,0,0,0,0,336,0,0,0,290,1] >;
C7⋊D12 in GAP, Magma, Sage, TeX
C_7\rtimes D_{12} % in TeX
G:=Group("C7:D12"); // GroupNames label
G:=SmallGroup(168,17);
// by ID
G=gap.SmallGroup(168,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,20,61,168,3604]);
// Polycyclic
G:=Group<a,b,c|a^7=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C7⋊D12 in TeX
Character table of C7⋊D12 in TeX