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## G = Dic3×D7order 168 = 23·3·7

### Direct product of Dic3 and D7

Aliases: Dic3×D7, D14.S3, C6.1D14, C14.1D6, Dic212C2, C42.1C22, (C3×D7)⋊C4, C33(C4×D7), C211(C2×C4), (C6×D7).C2, C2.1(S3×D7), C71(C2×Dic3), (C7×Dic3)⋊1C2, SmallGroup(168,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — Dic3×D7
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7
 Lower central C21 — Dic3×D7
 Upper central C1 — C2

Generators and relations for Dic3×D7
G = < a,b,c,d | a6=c7=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of Dic3×D7

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 7A 7B 7C 14A 14B 14C 21A 21B 21C 28A 28B 28C 28D 28E 28F 42A 42B 42C size 1 1 7 7 2 3 3 21 21 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 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 -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 1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 -i i i -i -1 1 -1 1 1 1 -1 -1 -1 1 1 1 i -i i -i -i i -1 -1 -1 linear of order 4 ρ6 1 -1 -1 1 1 i -i i -i -1 -1 1 1 1 1 -1 -1 -1 1 1 1 -i i -i i i -i -1 -1 -1 linear of order 4 ρ7 1 -1 1 -1 1 i -i -i i -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -i i -i i i -i -1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 -i i -i i -1 -1 1 1 1 1 -1 -1 -1 1 1 1 i -i i -i -i i -1 -1 -1 linear of order 4 ρ9 2 2 0 0 2 2 2 0 0 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D7 ρ10 2 2 -2 -2 -1 0 0 0 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 ρ11 2 2 2 2 -1 0 0 0 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 ρ12 2 2 0 0 2 -2 -2 0 0 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D14 ρ13 2 2 0 0 2 -2 -2 0 0 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D14 ρ14 2 2 0 0 2 2 2 0 0 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D7 ρ15 2 2 0 0 2 -2 -2 0 0 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D14 ρ16 2 2 0 0 2 2 2 0 0 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D7 ρ17 2 -2 -2 2 -1 0 0 0 0 1 1 -1 2 2 2 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ18 2 -2 2 -2 -1 0 0 0 0 1 -1 1 2 2 2 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ19 2 -2 0 0 2 2i -2i 0 0 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ43ζ76+ζ43ζ7 ζ4ζ75+ζ4ζ72 ζ43ζ75+ζ43ζ72 ζ4ζ74+ζ4ζ73 ζ4ζ76+ζ4ζ7 ζ43ζ74+ζ43ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 complex lifted from C4×D7 ρ20 2 -2 0 0 2 2i -2i 0 0 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ43ζ74+ζ43ζ73 ζ4ζ76+ζ4ζ7 ζ43ζ76+ζ43ζ7 ζ4ζ75+ζ4ζ72 ζ4ζ74+ζ4ζ73 ζ43ζ75+ζ43ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 complex lifted from C4×D7 ρ21 2 -2 0 0 2 2i -2i 0 0 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ43ζ75+ζ43ζ72 ζ4ζ74+ζ4ζ73 ζ43ζ74+ζ43ζ73 ζ4ζ76+ζ4ζ7 ζ4ζ75+ζ4ζ72 ζ43ζ76+ζ43ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 complex lifted from C4×D7 ρ22 2 -2 0 0 2 -2i 2i 0 0 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ4ζ76+ζ4ζ7 ζ43ζ75+ζ43ζ72 ζ4ζ75+ζ4ζ72 ζ43ζ74+ζ43ζ73 ζ43ζ76+ζ43ζ7 ζ4ζ74+ζ4ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 complex lifted from C4×D7 ρ23 2 -2 0 0 2 -2i 2i 0 0 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ4ζ75+ζ4ζ72 ζ43ζ74+ζ43ζ73 ζ4ζ74+ζ4ζ73 ζ43ζ76+ζ43ζ7 ζ43ζ75+ζ43ζ72 ζ4ζ76+ζ4ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 complex lifted from C4×D7 ρ24 2 -2 0 0 2 -2i 2i 0 0 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ4ζ74+ζ4ζ73 ζ43ζ76+ζ43ζ7 ζ4ζ76+ζ4ζ7 ζ43ζ75+ζ43ζ72 ζ43ζ74+ζ43ζ73 ζ4ζ75+ζ4ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 complex lifted from C4×D7 ρ25 4 4 0 0 -2 0 0 0 0 -2 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 0 0 0 0 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 orthogonal lifted from S3×D7 ρ26 4 4 0 0 -2 0 0 0 0 -2 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 0 0 0 0 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 orthogonal lifted from S3×D7 ρ27 4 4 0 0 -2 0 0 0 0 -2 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 0 0 0 0 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 orthogonal lifted from S3×D7 ρ28 4 -4 0 0 -2 0 0 0 0 2 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 0 0 0 0 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 -2 0 0 0 0 2 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 0 0 0 0 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 symplectic faithful, Schur index 2 ρ30 4 -4 0 0 -2 0 0 0 0 2 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 0 0 0 0 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 symplectic faithful, Schur index 2

Smallest permutation representation of Dic3×D7
On 84 points
Generators in S84
(1 41 13 27 20 34)(2 42 14 28 21 35)(3 36 8 22 15 29)(4 37 9 23 16 30)(5 38 10 24 17 31)(6 39 11 25 18 32)(7 40 12 26 19 33)(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 69 27 48)(2 70 28 49)(3 64 22 43)(4 65 23 44)(5 66 24 45)(6 67 25 46)(7 68 26 47)(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)

G:=sub<Sym(84)| (1,41,13,27,20,34)(2,42,14,28,21,35)(3,36,8,22,15,29)(4,37,9,23,16,30)(5,38,10,24,17,31)(6,39,11,25,18,32)(7,40,12,26,19,33)(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), (1,69,27,48)(2,70,28,49)(3,64,22,43)(4,65,23,44)(5,66,24,45)(6,67,25,46)(7,68,26,47)(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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)>;

G:=Group( (1,41,13,27,20,34)(2,42,14,28,21,35)(3,36,8,22,15,29)(4,37,9,23,16,30)(5,38,10,24,17,31)(6,39,11,25,18,32)(7,40,12,26,19,33)(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), (1,69,27,48)(2,70,28,49)(3,64,22,43)(4,65,23,44)(5,66,24,45)(6,67,25,46)(7,68,26,47)(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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83) );

G=PermutationGroup([[(1,41,13,27,20,34),(2,42,14,28,21,35),(3,36,8,22,15,29),(4,37,9,23,16,30),(5,38,10,24,17,31),(6,39,11,25,18,32),(7,40,12,26,19,33),(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)], [(1,69,27,48),(2,70,28,49),(3,64,22,43),(4,65,23,44),(5,66,24,45),(6,67,25,46),(7,68,26,47),(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,45),(46,49),(47,48),(50,52),(53,56),(54,55),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83)]])

Dic3×D7 is a maximal subgroup of   D285S3  D28⋊S3  C4×S3×D7  Dic7.D6  C42.C23
Dic3×D7 is a maximal quotient of   C28.32D6  D14⋊Dic3  C42.Q8

Matrix representation of Dic3×D7 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 1 336 0 0 1 0
,
 336 0 0 0 0 336 0 0 0 0 310 133 0 0 106 27
,
 0 1 0 0 336 143 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,336,0],[336,0,0,0,0,336,0,0,0,0,310,106,0,0,133,27],[0,336,0,0,1,143,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

Dic3×D7 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times D_7
% in TeX

G:=Group("Dic3xD7");
// GroupNames label

G:=SmallGroup(168,12);
// by ID

G=gap.SmallGroup(168,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,26,168,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^7=d^2=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=c^-1>;
// generators/relations

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