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## G = S3×Dic7order 168 = 23·3·7

### Direct product of S3 and Dic7

Aliases: S3×Dic7, D6.D7, C14.2D6, C6.2D14, Dic213C2, C42.2C22, (S3×C7)⋊C4, C73(C4×S3), C212(C2×C4), (S3×C14).C2, C2.2(S3×D7), C31(C2×Dic7), (C3×Dic7)⋊1C2, SmallGroup(168,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — S3×Dic7
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — S3×Dic7
 Lower central C21 — S3×Dic7
 Upper central C1 — C2

Generators and relations for S3×Dic7
G = < a,b,c,d | a3=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Character table of S3×Dic7

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 7A 7B 7C 12A 12B 14A 14B 14C 14D 14E 14F 14G 14H 14I 21A 21B 21C 42A 42B 42C size 1 1 3 3 2 7 7 21 21 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 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 -i i -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 4 ρ6 1 -1 1 -1 1 i -i -i i -1 1 1 1 i -i -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 4 ρ7 1 -1 -1 1 1 i -i i -i -1 1 1 1 i -i -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 -i i -i i -1 1 1 1 -i i -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 linear of order 4 ρ9 2 2 0 0 -1 2 2 0 0 -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 ρ10 2 2 0 0 -1 -2 -2 0 0 -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 ρ11 2 2 -2 -2 2 0 0 0 0 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ12 2 2 -2 -2 2 0 0 0 0 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ13 2 2 -2 -2 2 0 0 0 0 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ14 2 2 2 2 2 0 0 0 0 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ15 2 2 2 2 2 0 0 0 0 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ16 2 2 2 2 2 0 0 0 0 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ17 2 -2 2 -2 2 0 0 0 0 -2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 symplectic lifted from Dic7, Schur index 2 ρ18 2 -2 2 -2 2 0 0 0 0 -2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 symplectic lifted from Dic7, Schur index 2 ρ19 2 -2 -2 2 2 0 0 0 0 -2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 symplectic lifted from Dic7, Schur index 2 ρ20 2 -2 -2 2 2 0 0 0 0 -2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 symplectic lifted from Dic7, Schur index 2 ρ21 2 -2 2 -2 2 0 0 0 0 -2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 symplectic lifted from Dic7, Schur index 2 ρ22 2 -2 -2 2 2 0 0 0 0 -2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 symplectic lifted from Dic7, Schur index 2 ρ23 2 -2 0 0 -1 -2i 2i 0 0 1 2 2 2 i -i -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 complex lifted from C4×S3 ρ24 2 -2 0 0 -1 2i -2i 0 0 1 2 2 2 -i i -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 complex lifted from C4×S3 ρ25 4 4 0 0 -2 0 0 0 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 ρ26 4 4 0 0 -2 0 0 0 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 ρ27 4 4 0 0 -2 0 0 0 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 ρ28 4 -4 0 0 -2 0 0 0 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 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 -2 0 0 0 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 symplectic faithful, Schur index 2 ρ30 4 -4 0 0 -2 0 0 0 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 symplectic faithful, Schur index 2

Smallest permutation representation of S3×Dic7
On 84 points
Generators in S84
(1 17 53)(2 18 54)(3 19 55)(4 20 56)(5 21 43)(6 22 44)(7 23 45)(8 24 46)(9 25 47)(10 26 48)(11 27 49)(12 28 50)(13 15 51)(14 16 52)(29 71 70)(30 72 57)(31 73 58)(32 74 59)(33 75 60)(34 76 61)(35 77 62)(36 78 63)(37 79 64)(38 80 65)(39 81 66)(40 82 67)(41 83 68)(42 84 69)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 43)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
(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 69 8 62)(2 68 9 61)(3 67 10 60)(4 66 11 59)(5 65 12 58)(6 64 13 57)(7 63 14 70)(15 30 22 37)(16 29 23 36)(17 42 24 35)(18 41 25 34)(19 40 26 33)(20 39 27 32)(21 38 28 31)(43 80 50 73)(44 79 51 72)(45 78 52 71)(46 77 53 84)(47 76 54 83)(48 75 55 82)(49 74 56 81)

G:=sub<Sym(84)| (1,17,53)(2,18,54)(3,19,55)(4,20,56)(5,21,43)(6,22,44)(7,23,45)(8,24,46)(9,25,47)(10,26,48)(11,27,49)(12,28,50)(13,15,51)(14,16,52)(29,71,70)(30,72,57)(31,73,58)(32,74,59)(33,75,60)(34,76,61)(35,77,62)(36,78,63)(37,79,64)(38,80,65)(39,81,66)(40,82,67)(41,83,68)(42,84,69), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,43)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (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), (1,69,8,62)(2,68,9,61)(3,67,10,60)(4,66,11,59)(5,65,12,58)(6,64,13,57)(7,63,14,70)(15,30,22,37)(16,29,23,36)(17,42,24,35)(18,41,25,34)(19,40,26,33)(20,39,27,32)(21,38,28,31)(43,80,50,73)(44,79,51,72)(45,78,52,71)(46,77,53,84)(47,76,54,83)(48,75,55,82)(49,74,56,81)>;

G:=Group( (1,17,53)(2,18,54)(3,19,55)(4,20,56)(5,21,43)(6,22,44)(7,23,45)(8,24,46)(9,25,47)(10,26,48)(11,27,49)(12,28,50)(13,15,51)(14,16,52)(29,71,70)(30,72,57)(31,73,58)(32,74,59)(33,75,60)(34,76,61)(35,77,62)(36,78,63)(37,79,64)(38,80,65)(39,81,66)(40,82,67)(41,83,68)(42,84,69), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,43)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (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), (1,69,8,62)(2,68,9,61)(3,67,10,60)(4,66,11,59)(5,65,12,58)(6,64,13,57)(7,63,14,70)(15,30,22,37)(16,29,23,36)(17,42,24,35)(18,41,25,34)(19,40,26,33)(20,39,27,32)(21,38,28,31)(43,80,50,73)(44,79,51,72)(45,78,52,71)(46,77,53,84)(47,76,54,83)(48,75,55,82)(49,74,56,81) );

G=PermutationGroup([(1,17,53),(2,18,54),(3,19,55),(4,20,56),(5,21,43),(6,22,44),(7,23,45),(8,24,46),(9,25,47),(10,26,48),(11,27,49),(12,28,50),(13,15,51),(14,16,52),(29,71,70),(30,72,57),(31,73,58),(32,74,59),(33,75,60),(34,76,61),(35,77,62),(36,78,63),(37,79,64),(38,80,65),(39,81,66),(40,82,67),(41,83,68),(42,84,69)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,43),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70)], [(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)], [(1,69,8,62),(2,68,9,61),(3,67,10,60),(4,66,11,59),(5,65,12,58),(6,64,13,57),(7,63,14,70),(15,30,22,37),(16,29,23,36),(17,42,24,35),(18,41,25,34),(19,40,26,33),(20,39,27,32),(21,38,28,31),(43,80,50,73),(44,79,51,72),(45,78,52,71),(46,77,53,84),(47,76,54,83),(48,75,55,82),(49,74,56,81)])

S3×Dic7 is a maximal subgroup of   D12⋊D7  D125D7  C4×S3×D7  C42.C23  Dic3.D14
S3×Dic7 is a maximal quotient of   D6.Dic7  D6⋊Dic7  C14.Dic6

Matrix representation of S3×Dic7 in GL5(𝔽337)

 1 0 0 0 0 0 336 336 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 336 336 0 0 0 0 0 1 0 0 0 0 0 1
,
 336 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 109 1 0 0 0 335 34
,
 148 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 310 263 0 0 0 192 27

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,336,1,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,336,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1],[336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,109,335,0,0,0,1,34],[148,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,310,192,0,0,0,263,27] >;

S3×Dic7 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_7
% in TeX

G:=Group("S3xDic7");
// GroupNames label

G:=SmallGroup(168,13);
// by ID

G=gap.SmallGroup(168,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,168,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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