direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×D7, D35⋊C2, C5⋊1D14, C7⋊1D10, C35⋊C22, (C7×D5)⋊C2, (C5×D7)⋊C2, SmallGroup(140,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — D5×D7 |
Generators and relations for D5×D7
G = < a,b,c,d | a5=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of D5×D7
| class | 1 | 2A | 2B | 2C | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 35A | 35B | 35C | 35D | 35E | 35F | |
| size | 1 | 5 | 7 | 35 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 10 | 10 | 10 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 0 | -2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ7 | 2 | 0 | -2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ8 | 2 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ15 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | 0 | 0 | ζ54ζ74+ζ54ζ73+ζ5ζ74+ζ5ζ73 | ζ53ζ76+ζ53ζ7+ζ52ζ76+ζ52ζ7 | ζ53ζ75+ζ53ζ72+ζ52ζ75+ζ52ζ72 | ζ54ζ76+ζ54ζ7+ζ5ζ76+ζ5ζ7 | ζ54ζ75+ζ54ζ72+ζ5ζ75+ζ5ζ72 | ζ53ζ74+ζ53ζ73+ζ52ζ74+ζ52ζ73 | orthogonal faithful |
| ρ16 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | 0 | 0 | ζ53ζ76+ζ53ζ7+ζ52ζ76+ζ52ζ7 | ζ54ζ75+ζ54ζ72+ζ5ζ75+ζ5ζ72 | ζ54ζ74+ζ54ζ73+ζ5ζ74+ζ5ζ73 | ζ53ζ75+ζ53ζ72+ζ52ζ75+ζ52ζ72 | ζ53ζ74+ζ53ζ73+ζ52ζ74+ζ52ζ73 | ζ54ζ76+ζ54ζ7+ζ5ζ76+ζ5ζ7 | orthogonal faithful |
| ρ17 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | 0 | 0 | ζ53ζ75+ζ53ζ72+ζ52ζ75+ζ52ζ72 | ζ54ζ74+ζ54ζ73+ζ5ζ74+ζ5ζ73 | ζ54ζ76+ζ54ζ7+ζ5ζ76+ζ5ζ7 | ζ53ζ74+ζ53ζ73+ζ52ζ74+ζ52ζ73 | ζ53ζ76+ζ53ζ7+ζ52ζ76+ζ52ζ7 | ζ54ζ75+ζ54ζ72+ζ5ζ75+ζ5ζ72 | orthogonal faithful |
| ρ18 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | 0 | 0 | ζ54ζ76+ζ54ζ7+ζ5ζ76+ζ5ζ7 | ζ53ζ75+ζ53ζ72+ζ52ζ75+ζ52ζ72 | ζ53ζ74+ζ53ζ73+ζ52ζ74+ζ52ζ73 | ζ54ζ75+ζ54ζ72+ζ5ζ75+ζ5ζ72 | ζ54ζ74+ζ54ζ73+ζ5ζ74+ζ5ζ73 | ζ53ζ76+ζ53ζ7+ζ52ζ76+ζ52ζ7 | orthogonal faithful |
| ρ19 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | 0 | 0 | ζ53ζ74+ζ53ζ73+ζ52ζ74+ζ52ζ73 | ζ54ζ76+ζ54ζ7+ζ5ζ76+ζ5ζ7 | ζ54ζ75+ζ54ζ72+ζ5ζ75+ζ5ζ72 | ζ53ζ76+ζ53ζ7+ζ52ζ76+ζ52ζ7 | ζ53ζ75+ζ53ζ72+ζ52ζ75+ζ52ζ72 | ζ54ζ74+ζ54ζ73+ζ5ζ74+ζ5ζ73 | orthogonal faithful |
| ρ20 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | 0 | 0 | ζ54ζ75+ζ54ζ72+ζ5ζ75+ζ5ζ72 | ζ53ζ74+ζ53ζ73+ζ52ζ74+ζ52ζ73 | ζ53ζ76+ζ53ζ7+ζ52ζ76+ζ52ζ7 | ζ54ζ74+ζ54ζ73+ζ5ζ74+ζ5ζ73 | ζ54ζ76+ζ54ζ7+ζ5ζ76+ζ5ζ7 | ζ53ζ75+ζ53ζ72+ζ52ζ75+ζ52ζ72 | orthogonal faithful |
►On 35 points
Generators in S35
(1 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)
(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)
(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)
G:=sub<Sym(35)| (1,34,27,20,13)(2,35,28,21,14)(3,29,22,15,8)(4,30,23,16,9)(5,31,24,17,10)(6,32,25,18,11)(7,33,26,19,12), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (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)>;
G:=Group( (1,34,27,20,13)(2,35,28,21,14)(3,29,22,15,8)(4,30,23,16,9)(5,31,24,17,10)(6,32,25,18,11)(7,33,26,19,12), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (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) );
G=PermutationGroup([[(1,34,27,20,13),(2,35,28,21,14),(3,29,22,15,8),(4,30,23,16,9),(5,31,24,17,10),(6,32,25,18,11),(7,33,26,19,12)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35)], [(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)], [(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)]])
D5×D7 is a maximal subgroup of
D15⋊D7
D5×D7 is a maximal quotient of D70.C2 C35⋊D4 C5⋊D28 C7⋊D20 C35⋊Q8 D15⋊D7
Matrix representation of D5×D7 ►in GL4(𝔽71) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 7 |
| 0 | 0 | 35 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 0 |
| 0 | 0 | 67 | 1 |
| 70 | 1 | 0 | 0 |
| 55 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 70 | 0 | 0 | 0 |
| 55 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(71))| [1,0,0,0,0,1,0,0,0,0,61,35,0,0,7,18],[1,0,0,0,0,1,0,0,0,0,70,67,0,0,0,1],[70,55,0,0,1,15,0,0,0,0,1,0,0,0,0,1],[70,55,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;
D5×D7 in GAP, Magma, Sage, TeX
D_5\times D_7
% in TeX
G:=Group("D5xD7"); // GroupNames label
G:=SmallGroup(140,7);
// by ID
G=gap.SmallGroup(140,7);
# by ID
G:=PCGroup([4,-2,-2,-5,-7,102,1923]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^7=d^2=1,b*a*b=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
Export
Subgroup lattice of D5×D7 in TeX
Character table of D5×D7 in TeX