direct product, metabelian, supersoluble, monomial, A-group
Aliases: D7×C3⋊S3, C21⋊1D6, C32⋊4D14, (C3×D7)⋊S3, C3⋊2(S3×D7), C3⋊D21⋊1C2, (C3×C21)⋊2C22, (C32×D7)⋊2C2, C7⋊1(C2×C3⋊S3), (C7×C3⋊S3)⋊1C2, SmallGroup(252,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — D7×C3⋊S3 |
Generators and relations for D7×C3⋊S3
G = < a,b,c,d,e | a7=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 456 in 60 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C22, S3, C6, C7, C32, D6, D7, D7, C14, C3⋊S3, C3⋊S3, C3×C6, C21, D14, C2×C3⋊S3, S3×C7, C3×D7, D21, C3×C21, S3×D7, C32×D7, C7×C3⋊S3, C3⋊D21, D7×C3⋊S3
Quotients: C1, C2, C22, S3, D6, D7, C3⋊S3, D14, C2×C3⋊S3, S3×D7, D7×C3⋊S3
Character table of D7×C3⋊S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | 21G | 21H | 21I | 21J | 21K | 21L | |
| size | 1 | 7 | 9 | 63 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 18 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 1 | 1 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from D6 |
| ρ13 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ14 | 2 | 0 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ15 | 2 | 0 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ16 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ17 | 2 | 0 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ18 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ19 | 4 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ20 | 4 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | 2ζ76+2ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | orthogonal lifted from S3×D7 |
| ρ21 | 4 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | 2ζ74+2ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | orthogonal lifted from S3×D7 |
| ρ22 | 4 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ75-ζ72 | 2ζ75+2ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ23 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ76-ζ7 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 2ζ74+2ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ24 | 4 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ25 | 4 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | 2ζ75+2ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | orthogonal lifted from S3×D7 |
| ρ26 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ75-ζ72 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 2ζ76+2ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ27 | 4 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ76-ζ7 | 2ζ76+2ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ28 | 4 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ74-ζ73 | 2ζ74+2ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ29 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ74-ζ73 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 2ζ75+2ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ30 | 4 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
►On 63 points
Generators in S63
(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)
(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)
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)
(1 27 48)(2 28 49)(3 22 43)(4 23 44)(5 24 45)(6 25 46)(7 26 47)(8 29 50)(9 30 51)(10 31 52)(11 32 53)(12 33 54)(13 34 55)(14 35 56)(15 36 57)(16 37 58)(17 38 59)(18 39 60)(19 40 61)(20 41 62)(21 42 63)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
G:=sub<Sym(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), (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), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (1,27,48)(2,28,49)(3,22,43)(4,23,44)(5,24,45)(6,25,46)(7,26,47)(8,29,50)(9,30,51)(10,31,52)(11,32,53)(12,33,54)(13,34,55)(14,35,56)(15,36,57)(16,37,58)(17,38,59)(18,39,60)(19,40,61)(20,41,62)(21,42,63), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)>;
G:=Group( (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), (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), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (1,27,48)(2,28,49)(3,22,43)(4,23,44)(5,24,45)(6,25,46)(7,26,47)(8,29,50)(9,30,51)(10,31,52)(11,32,53)(12,33,54)(13,34,55)(14,35,56)(15,36,57)(16,37,58)(17,38,59)(18,39,60)(19,40,61)(20,41,62)(21,42,63), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56) );
G=PermutationGroup([[(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)], [(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)], [(1,20,13),(2,21,14),(3,15,8),(4,16,9),(5,17,10),(6,18,11),(7,19,12),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56)], [(1,27,48),(2,28,49),(3,22,43),(4,23,44),(5,24,45),(6,25,46),(7,26,47),(8,29,50),(9,30,51),(10,31,52),(11,32,53),(12,33,54),(13,34,55),(14,35,56),(15,36,57),(16,37,58),(17,38,59),(18,39,60),(19,40,61),(20,41,62),(21,42,63)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)]])
Matrix representation of D7×C3⋊S3 ►in GL6(𝔽43)
| 42 | 1 | 0 | 0 | 0 | 0 |
| 22 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 22 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 42 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 42 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 42 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 42 |
G:=sub<GL(6,GF(43))| [42,22,0,0,0,0,1,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,22,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42,0,0,0,0,1,42,0,0,0,0,0,0,0,42,0,0,0,0,1,42],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,1,0,0,0,0,42,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,42,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1,42] >;
D7×C3⋊S3 in GAP, Magma, Sage, TeX
D_7\times C_3\rtimes S_3
% in TeX
G:=Group("D7xC3:S3"); // GroupNames label
G:=SmallGroup(252,34);
// by ID
G=gap.SmallGroup(252,34);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,67,248,5404]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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