direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×Dic7, D6.D7, C14.2D6, C6.2D14, Dic21⋊3C2, C42.2C22, (S3×C7)⋊C4, C7⋊3(C4×S3), C21⋊2(C2×C4), (S3×C14).C2, C2.2(S3×D7), C3⋊1(C2×Dic7), (C3×Dic7)⋊1C2, SmallGroup(168,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — S3×Dic7 |
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 |
►On 84 points
Generators in S84
(1 41 82)(2 42 83)(3 29 84)(4 30 71)(5 31 72)(6 32 73)(7 33 74)(8 34 75)(9 35 76)(10 36 77)(11 37 78)(12 38 79)(13 39 80)(14 40 81)(15 59 52)(16 60 53)(17 61 54)(18 62 55)(19 63 56)(20 64 43)(21 65 44)(22 66 45)(23 67 46)(24 68 47)(25 69 48)(26 70 49)(27 57 50)(28 58 51)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 66)(16 67)(17 68)(18 69)(19 70)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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 50 8 43)(2 49 9 56)(3 48 10 55)(4 47 11 54)(5 46 12 53)(6 45 13 52)(7 44 14 51)(15 32 22 39)(16 31 23 38)(17 30 24 37)(18 29 25 36)(19 42 26 35)(20 41 27 34)(21 40 28 33)(57 75 64 82)(58 74 65 81)(59 73 66 80)(60 72 67 79)(61 71 68 78)(62 84 69 77)(63 83 70 76)
G:=sub<Sym(84)| (1,41,82)(2,42,83)(3,29,84)(4,30,71)(5,31,72)(6,32,73)(7,33,74)(8,34,75)(9,35,76)(10,36,77)(11,37,78)(12,38,79)(13,39,80)(14,40,81)(15,59,52)(16,60,53)(17,61,54)(18,62,55)(19,63,56)(20,64,43)(21,65,44)(22,66,45)(23,67,46)(24,68,47)(25,69,48)(26,70,49)(27,57,50)(28,58,51), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,66)(16,67)(17,68)(18,69)(19,70)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,50,8,43)(2,49,9,56)(3,48,10,55)(4,47,11,54)(5,46,12,53)(6,45,13,52)(7,44,14,51)(15,32,22,39)(16,31,23,38)(17,30,24,37)(18,29,25,36)(19,42,26,35)(20,41,27,34)(21,40,28,33)(57,75,64,82)(58,74,65,81)(59,73,66,80)(60,72,67,79)(61,71,68,78)(62,84,69,77)(63,83,70,76)>;
G:=Group( (1,41,82)(2,42,83)(3,29,84)(4,30,71)(5,31,72)(6,32,73)(7,33,74)(8,34,75)(9,35,76)(10,36,77)(11,37,78)(12,38,79)(13,39,80)(14,40,81)(15,59,52)(16,60,53)(17,61,54)(18,62,55)(19,63,56)(20,64,43)(21,65,44)(22,66,45)(23,67,46)(24,68,47)(25,69,48)(26,70,49)(27,57,50)(28,58,51), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,66)(16,67)(17,68)(18,69)(19,70)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,50,8,43)(2,49,9,56)(3,48,10,55)(4,47,11,54)(5,46,12,53)(6,45,13,52)(7,44,14,51)(15,32,22,39)(16,31,23,38)(17,30,24,37)(18,29,25,36)(19,42,26,35)(20,41,27,34)(21,40,28,33)(57,75,64,82)(58,74,65,81)(59,73,66,80)(60,72,67,79)(61,71,68,78)(62,84,69,77)(63,83,70,76) );
G=PermutationGroup([[(1,41,82),(2,42,83),(3,29,84),(4,30,71),(5,31,72),(6,32,73),(7,33,74),(8,34,75),(9,35,76),(10,36,77),(11,37,78),(12,38,79),(13,39,80),(14,40,81),(15,59,52),(16,60,53),(17,61,54),(18,62,55),(19,63,56),(20,64,43),(21,65,44),(22,66,45),(23,67,46),(24,68,47),(25,69,48),(26,70,49),(27,57,50),(28,58,51)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,66),(16,67),(17,68),(18,69),(19,70),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,71),(38,72),(39,73),(40,74),(41,75),(42,76),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,50,8,43),(2,49,9,56),(3,48,10,55),(4,47,11,54),(5,46,12,53),(6,45,13,52),(7,44,14,51),(15,32,22,39),(16,31,23,38),(17,30,24,37),(18,29,25,36),(19,42,26,35),(20,41,27,34),(21,40,28,33),(57,75,64,82),(58,74,65,81),(59,73,66,80),(60,72,67,79),(61,71,68,78),(62,84,69,77),(63,83,70,76)]])
S3×Dic7 is a maximal subgroup of
D12⋊D7 D12⋊5D7 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
Export
Subgroup lattice of S3×Dic7 in TeX
Character table of S3×Dic7 in TeX