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## G = C5⋊F7order 210 = 2·3·5·7

### The semidirect product of C5 and F7 acting via F7/C7⋊C3=C2

Aliases: C5⋊F7, D35⋊C3, C351C6, C7⋊C3⋊D5, C7⋊(C3×D5), (C5×C7⋊C3)⋊1C2, SmallGroup(210,3)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5⋊F7
G = < a,b,c | a5=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C5⋊F7

 class 1 2 3A 3B 5A 5B 6A 6B 7 15A 15B 15C 15D 35A 35B 35C 35D size 1 35 7 7 2 2 35 35 6 14 14 14 14 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 ζ32 ζ3 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 linear of order 3 ρ4 1 -1 ζ32 ζ3 1 1 ζ65 ζ6 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 linear of order 6 ρ5 1 1 ζ3 ζ32 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 linear of order 3 ρ6 1 -1 ζ3 ζ32 1 1 ζ6 ζ65 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 linear of order 6 ρ7 2 0 2 2 -1-√5/2 -1+√5/2 0 0 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 ρ8 2 0 2 2 -1+√5/2 -1-√5/2 0 0 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 ρ9 2 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 0 0 2 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 complex lifted from C3×D5 ρ10 2 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 0 0 2 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 complex lifted from C3×D5 ρ11 2 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 0 0 2 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 complex lifted from C3×D5 ρ12 2 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 0 0 2 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 complex lifted from C3×D5 ρ13 6 0 0 0 6 6 0 0 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F7 ρ14 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -1 0 0 0 0 -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 orthogonal faithful ρ15 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -1 0 0 0 0 ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 orthogonal faithful ρ16 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -1 0 0 0 0 ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 orthogonal faithful ρ17 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -1 0 0 0 0 -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 orthogonal faithful

Smallest permutation representation of C5⋊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 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)(8 29)(9 32 10 35 12 34)(11 31 14 33 13 30)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)

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

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

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

C5⋊F7 is a maximal subgroup of   D5×F7
C5⋊F7 is a maximal quotient of   C353C12

Matrix representation of C5⋊F7 in GL6(𝔽211)

 104 0 181 30 181 0 30 104 181 0 0 181 30 30 74 0 181 0 0 30 0 104 181 181 30 0 0 30 74 181 0 30 181 30 0 74
,
 0 0 0 0 0 210 1 0 0 0 0 210 0 1 0 0 0 210 0 0 1 0 0 210 0 0 0 1 0 210 0 0 0 0 1 210
,
 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0

G:=sub<GL(6,GF(211))| [104,30,30,0,30,0,0,104,30,30,0,30,181,181,74,0,0,181,30,0,0,104,30,30,181,0,181,181,74,0,0,181,0,181,181,74],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,210,210,210,210,210,210],[0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C5⋊F7 in GAP, Magma, Sage, TeX

C_5\rtimes F_7
% in TeX

G:=Group("C5:F7");
// GroupNames label

G:=SmallGroup(210,3);
// by ID

G=gap.SmallGroup(210,3);
# by ID

G:=PCGroup([4,-2,-3,-5,-7,290,2883,487]);
// Polycyclic

G:=Group<a,b,c|a^5=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

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