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## G = D7×F5order 280 = 23·5·7

### Direct product of D7 and F5

Aliases: D7×F5, D35⋊C4, D5.1D14, C5⋊(C4×D7), C7⋊F5⋊C2, C35⋊(C2×C4), (C5×D7)⋊C4, (C7×F5)⋊C2, C71(C2×F5), (D5×D7).C2, (C7×D5).C22, SmallGroup(280,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — D7×F5
 Chief series C1 — C7 — C35 — C7×D5 — D5×D7 — D7×F5
 Lower central C35 — D7×F5
 Upper central C1

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

Smallest permutation representation of D7×F5
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

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