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## G = C3⋊F7order 126 = 2·32·7

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

Aliases: C3⋊F7, D21⋊C3, C211C6, C7⋊C3⋊S3, C7⋊(C3×S3), (C3×C7⋊C3)⋊1C2, SmallGroup(126,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C3⋊F7
 Chief series C1 — C7 — C21 — C3×C7⋊C3 — C3⋊F7
 Lower central C21 — C3⋊F7
 Upper central C1

Generators and relations for C3⋊F7
G = < a,b,c | a3=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C3⋊F7

 class 1 2 3A 3B 3C 3D 3E 6A 6B 7 21A 21B size 1 21 2 7 7 14 14 21 21 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 linear of order 3 ρ4 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 linear of order 3 ρ5 1 -1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 1 linear of order 6 ρ6 1 -1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 1 linear of order 6 ρ7 2 0 -1 2 2 -1 -1 0 0 2 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 -1 -1 complex lifted from C3×S3 ρ9 2 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 -1 -1 complex lifted from C3×S3 ρ10 6 0 6 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ11 6 0 -3 0 0 0 0 0 0 -1 1+√21/2 1-√21/2 orthogonal faithful ρ12 6 0 -3 0 0 0 0 0 0 -1 1-√21/2 1+√21/2 orthogonal faithful

Permutation representations of C3⋊F7
On 21 points - transitive group 21T10
Generators in S21
```(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(2 4 3 7 5 6)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)```

`G:=sub<Sym(21)| (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)>;`

`G:=Group( (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16) );`

`G=PermutationGroup([(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(2,4,3,7,5,6),(8,15),(9,18,10,21,12,20),(11,17,14,19,13,16)])`

`G:=TransitiveGroup(21,10);`

C3⋊F7 is a maximal subgroup of   S3×F7  C9⋊F7  C92F7  C95F7  C32⋊F7  C324F7
C3⋊F7 is a maximal quotient of   C6.F7  C9⋊F7  C92F7  C95F7  D21⋊C9  C32⋊F7  C324F7

Matrix representation of C3⋊F7 in GL6(𝔽43)

 40 38 38 0 38 0 0 40 38 38 0 38 5 5 2 0 0 5 38 0 0 40 38 38 5 0 5 5 2 0 0 5 0 5 5 2
,
 42 42 42 42 42 42 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 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 42 42 42 42 42 42 0 0 0 0 1 0

`G:=sub<GL(6,GF(43))| [40,0,5,38,5,0,38,40,5,0,0,5,38,38,2,0,5,0,0,38,0,40,5,5,38,0,0,38,2,5,0,38,5,38,0,2],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[1,0,0,0,42,0,0,0,0,1,42,0,0,0,0,0,42,0,0,0,1,0,42,0,0,0,0,0,42,1,0,1,0,0,42,0] >;`

C3⋊F7 in GAP, Magma, Sage, TeX

`C_3\rtimes F_7`
`% in TeX`

`G:=Group("C3:F7");`
`// GroupNames label`

`G:=SmallGroup(126,9);`
`// by ID`

`G=gap.SmallGroup(126,9);`
`# by ID`

`G:=PCGroup([4,-2,-3,-3,-7,146,1731,295]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=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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