direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×F5, D35⋊C4, D5.1D14, C5⋊(C4×D7), C7⋊F5⋊C2, C35⋊(C2×C4), (C5×D7)⋊C4, (C7×F5)⋊C2, C7⋊1(C2×F5), (D5×D7).C2, (C7×D5).C22, SmallGroup(280,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — D7×F5 |
Generators and relations for D7×F5
G = < a,b,c,d | a7=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Character table of D7×F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 7A | 7B | 7C | 10 | 14A | 14B | 14C | 28A | 28B | 28C | 28D | 28E | 28F | 35A | 35B | 35C | |
| size | 1 | 5 | 7 | 35 | 5 | 5 | 35 | 35 | 4 | 2 | 2 | 2 | 28 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | -i | i | i | -i | 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 | i | -i | i | -i | 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 | i | -i | -i | i | 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 | -i | i | -i | i | 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 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ14 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ15 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ43ζ76+ζ43ζ7 | ζ4ζ76+ζ4ζ7 | ζ43ζ74+ζ43ζ73 | ζ43ζ75+ζ43ζ72 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | complex lifted from C4×D7 |
| ρ16 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ4ζ76+ζ4ζ7 | ζ43ζ76+ζ43ζ7 | ζ4ζ74+ζ4ζ73 | ζ4ζ75+ζ4ζ72 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | complex lifted from C4×D7 |
| ρ17 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ4ζ74+ζ4ζ73 | ζ43ζ74+ζ43ζ73 | ζ4ζ75+ζ4ζ72 | ζ4ζ76+ζ4ζ7 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | complex lifted from C4×D7 |
| ρ18 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ43ζ75+ζ43ζ72 | ζ4ζ75+ζ4ζ72 | ζ43ζ76+ζ43ζ7 | ζ43ζ74+ζ43ζ73 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | complex lifted from C4×D7 |
| ρ19 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ43ζ74+ζ43ζ73 | ζ4ζ74+ζ4ζ73 | ζ43ζ75+ζ43ζ72 | ζ43ζ76+ζ43ζ7 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | complex lifted from C4×D7 |
| ρ20 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ4ζ75+ζ4ζ72 | ζ43ζ75+ζ43ζ72 | ζ4ζ76+ζ4ζ7 | ζ4ζ74+ζ4ζ73 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | complex lifted from C4×D7 |
| ρ21 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ23 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4ζ76+4ζ7 | 4ζ74+4ζ73 | 4ζ75+4ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | orthogonal faithful |
| ρ24 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4ζ75+4ζ72 | 4ζ76+4ζ7 | 4ζ74+4ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | orthogonal faithful |
| ρ25 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4ζ74+4ζ73 | 4ζ75+4ζ72 | 4ζ76+4ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | orthogonal faithful |
►On 35 points
Generators in S35
(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)
(1 13 20 27 34)(2 14 21 28 35)(3 8 15 22 29)(4 9 16 23 30)(5 10 17 24 31)(6 11 18 25 32)(7 12 19 26 33)
(8 15 29 22)(9 16 30 23)(10 17 31 24)(11 18 32 25)(12 19 33 26)(13 20 34 27)(14 21 35 28)
G:=sub<Sym(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), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (8,15,29,22)(9,16,30,23)(10,17,31,24)(11,18,32,25)(12,19,33,26)(13,20,34,27)(14,21,35,28)>;
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), (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), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (8,15,29,22)(9,16,30,23)(10,17,31,24)(11,18,32,25)(12,19,33,26)(13,20,34,27)(14,21,35,28) );
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)], [(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)], [(1,13,20,27,34),(2,14,21,28,35),(3,8,15,22,29),(4,9,16,23,30),(5,10,17,24,31),(6,11,18,25,32),(7,12,19,26,33)], [(8,15,29,22),(9,16,30,23),(10,17,31,24),(11,18,32,25),(12,19,33,26),(13,20,34,27),(14,21,35,28)]])
Matrix representation of D7×F5 ►in GL6(𝔽281)
| 7 | 56 | 0 | 0 | 0 | 0 |
| 280 | 233 | 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 |
| 274 | 225 | 0 | 0 | 0 | 0 |
| 41 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 280 | 0 | 0 | 0 |
| 0 | 0 | 0 | 280 | 0 | 0 |
| 0 | 0 | 0 | 0 | 280 | 0 |
| 0 | 0 | 0 | 0 | 0 | 280 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 175 |
| 0 | 0 | 2 | 1 | 1 | 175 |
| 0 | 0 | 1 | 2 | 1 | 175 |
| 0 | 0 | 8 | 8 | 8 | 277 |
| 228 | 0 | 0 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 280 | 280 | 280 | 106 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(281))| [7,280,0,0,0,0,56,233,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[274,41,0,0,0,0,225,7,0,0,0,0,0,0,280,0,0,0,0,0,0,280,0,0,0,0,0,0,280,0,0,0,0,0,0,280],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,1,8,0,0,1,1,2,8,0,0,1,1,1,8,0,0,175,175,175,277],[228,0,0,0,0,0,0,228,0,0,0,0,0,0,0,1,280,0,0,0,0,0,280,0,0,0,1,0,280,0,0,0,0,0,106,1] >;
D7×F5 in GAP, Magma, Sage, TeX
D_7\times F_5
% in TeX
G:=Group("D7xF5"); // GroupNames label
G:=SmallGroup(280,32);
// by ID
G=gap.SmallGroup(280,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,26,168,173,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^5=d^4=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^-1=c^3>;
// generators/relations
Export
Subgroup lattice of D7×F5 in TeX
Character table of D7×F5 in TeX