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## G = D5×F7order 420 = 22·3·5·7

### Direct product of D5 and F7

Aliases: D5×F7, D35⋊C6, C5⋊F7⋊C2, C35⋊(C2×C6), (D5×D7)⋊C3, D7⋊(C3×D5), (C5×D7)⋊C6, (C7×D5)⋊C6, C7⋊C31D10, C71(C6×D5), (C5×F7)⋊C2, C51(C2×F7), (D5×C7⋊C3)⋊C2, (C5×C7⋊C3)⋊C22, SmallGroup(420,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — D5×F7
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — C5×F7 — D5×F7
 Lower central C35 — D5×F7
 Upper central C1

Generators and relations for D5×F7
G = < a,b,c,d | a5=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Character table of D5×F7

 class 1 2A 2B 2C 3A 3B 5A 5B 6A 6B 6C 6D 6E 6F 7 10A 10B 14 15A 15B 15C 15D 30A 30B 30C 30D 35A 35B size 1 5 7 35 7 7 2 2 7 7 35 35 35 35 6 14 14 30 14 14 14 14 14 14 14 14 12 12 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 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 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 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 linear of order 2 ρ5 1 -1 1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 1 1 1 -1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 linear of order 6 ρ6 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 linear of order 3 ρ7 1 1 -1 -1 ζ32 ζ3 1 1 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 -1 -1 1 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 ζ65 ζ6 1 1 linear of order 6 ρ8 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 linear of order 3 ρ9 1 -1 -1 1 ζ3 ζ32 1 1 ζ65 ζ6 ζ3 ζ65 ζ32 ζ6 1 -1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 ζ6 ζ65 1 1 linear of order 6 ρ10 1 1 -1 -1 ζ3 ζ32 1 1 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 -1 -1 1 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 ζ6 ζ65 1 1 linear of order 6 ρ11 1 -1 1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 1 1 1 -1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 linear of order 6 ρ12 1 -1 -1 1 ζ32 ζ3 1 1 ζ6 ζ65 ζ32 ζ6 ζ3 ζ65 1 -1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 ζ65 ζ6 1 1 linear of order 6 ρ13 2 0 -2 0 2 2 -1-√5/2 -1+√5/2 -2 -2 0 0 0 0 2 1-√5/2 1+√5/2 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ14 2 0 -2 0 2 2 -1+√5/2 -1-√5/2 -2 -2 0 0 0 0 2 1+√5/2 1-√5/2 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ15 2 0 2 0 2 2 -1-√5/2 -1+√5/2 2 2 0 0 0 0 2 -1+√5/2 -1-√5/2 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ16 2 0 2 0 2 2 -1+√5/2 -1-√5/2 2 2 0 0 0 0 2 -1-√5/2 -1+√5/2 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ17 2 0 -2 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 1-√-3 1+√-3 0 0 0 0 2 1+√5/2 1-√5/2 0 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 -ζ32ζ54-ζ32ζ5 -ζ3ζ54-ζ3ζ5 -ζ32ζ53-ζ32ζ52 -ζ3ζ53-ζ3ζ52 -1+√5/2 -1-√5/2 complex lifted from C6×D5 ρ18 2 0 -2 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 1-√-3 1+√-3 0 0 0 0 2 1-√5/2 1+√5/2 0 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 -ζ32ζ53-ζ32ζ52 -ζ3ζ53-ζ3ζ52 -ζ32ζ54-ζ32ζ5 -ζ3ζ54-ζ3ζ5 -1-√5/2 -1+√5/2 complex lifted from C6×D5 ρ19 2 0 2 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 -1-√-3 -1+√-3 0 0 0 0 2 -1-√5/2 -1+√5/2 0 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 -1+√5/2 -1-√5/2 complex lifted from C3×D5 ρ20 2 0 -2 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 1+√-3 1-√-3 0 0 0 0 2 1+√5/2 1-√5/2 0 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 -ζ3ζ54-ζ3ζ5 -ζ32ζ54-ζ32ζ5 -ζ3ζ53-ζ3ζ52 -ζ32ζ53-ζ32ζ52 -1+√5/2 -1-√5/2 complex lifted from C6×D5 ρ21 2 0 2 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 -1+√-3 -1-√-3 0 0 0 0 2 -1+√5/2 -1-√5/2 0 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 -1-√5/2 -1+√5/2 complex lifted from C3×D5 ρ22 2 0 2 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 -1-√-3 -1+√-3 0 0 0 0 2 -1+√5/2 -1-√5/2 0 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 -1-√5/2 -1+√5/2 complex lifted from C3×D5 ρ23 2 0 2 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 -1+√-3 -1-√-3 0 0 0 0 2 -1-√5/2 -1+√5/2 0 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 -1+√5/2 -1-√5/2 complex lifted from C3×D5 ρ24 2 0 -2 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 1+√-3 1-√-3 0 0 0 0 2 1-√5/2 1+√5/2 0 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 -ζ3ζ53-ζ3ζ52 -ζ32ζ53-ζ32ζ52 -ζ3ζ54-ζ3ζ5 -ζ32ζ54-ζ32ζ5 -1-√5/2 -1+√5/2 complex lifted from C6×D5 ρ25 6 -6 0 0 0 0 6 6 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from C2×F7 ρ26 6 6 0 0 0 0 6 6 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from F7 ρ27 12 0 0 0 0 0 -3+3√5 -3-3√5 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 1-√5/2 1+√5/2 orthogonal faithful ρ28 12 0 0 0 0 0 -3-3√5 -3+3√5 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 1+√5/2 1-√5/2 orthogonal faithful

Smallest permutation representation of D5×F7
On 35 points
Generators in S35
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(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)
(2 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 20)(23 25 24 28 26 27)(30 32 31 35 33 34)

G:=sub<Sym(35)| (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)(30,32,31,35,33,34)>;

G:=Group( (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)(30,32,31,35,33,34) );

G=PermutationGroup([[(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(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)], [(2,4,3,7,5,6),(9,11,10,14,12,13),(16,18,17,21,19,20),(23,25,24,28,26,27),(30,32,31,35,33,34)]])

Matrix representation of D5×F7 in GL8(𝔽211)

 210 1 0 0 0 0 0 0 31 179 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 180 210 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 210 210 210 210 210 210 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 210 210 210 210 210 210 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(211))| [210,31,0,0,0,0,0,0,1,179,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,180,0,0,0,0,0,0,0,210,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,210,1,0,0,0,0,0,0,210,0,1,0,0,0,0,0,210,0,0,1,0,0,0,0,210,0,0,0,1,0,0,0,210,0,0,0,0,1,0,0,210,0,0,0,0,0],[15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,210,0,0,0,0,0,0,1,210,0,0,0,0,0,0,0,210,0,0,0,0,0,1,0,210,0,0,0,0,0,0,0,210,1,0,0,0,1,0,0,210,0] >;

D5×F7 in GAP, Magma, Sage, TeX

D_5\times F_7
% in TeX

G:=Group("D5xF7");
// GroupNames label

G:=SmallGroup(420,16);
// by ID

G=gap.SmallGroup(420,16);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,488,9004,1514]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^7=d^6=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^5>;
// generators/relations

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