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 |
(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