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## G = C4⋊F7order 168 = 23·3·7

### The semidirect product of C4 and F7 acting via F7/C7⋊C3=C2

Aliases: C4⋊F7, D28⋊C3, C281C6, D141C6, C7⋊C31D4, C71(C3×D4), (C2×F7)⋊1C2, C2.4(C2×F7), C14.3(C2×C6), (C4×C7⋊C3)⋊1C2, (C2×C7⋊C3).3C22, SmallGroup(168,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4⋊F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4⋊F7
 Lower central C7 — C14 — C4⋊F7
 Upper central C1 — C2 — C4

Generators and relations for C4⋊F7
G = < a,b,c | a4=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C4⋊F7

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7 12A 12B 14 28A 28B size 1 1 14 14 7 7 2 7 7 14 14 14 14 6 14 14 6 6 6 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 1 1 linear of order 3 ρ6 1 1 -1 -1 ζ3 ζ32 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 1 ζ3 ζ32 1 1 1 linear of order 6 ρ7 1 1 1 -1 ζ3 ζ32 -1 ζ3 ζ32 ζ3 ζ6 ζ32 ζ65 1 ζ65 ζ6 1 -1 -1 linear of order 6 ρ8 1 1 1 -1 ζ32 ζ3 -1 ζ32 ζ3 ζ32 ζ65 ζ3 ζ6 1 ζ6 ζ65 1 -1 -1 linear of order 6 ρ9 1 1 -1 1 ζ3 ζ32 -1 ζ3 ζ32 ζ65 ζ32 ζ6 ζ3 1 ζ65 ζ6 1 -1 -1 linear of order 6 ρ10 1 1 -1 -1 ζ32 ζ3 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 1 ζ32 ζ3 1 1 1 linear of order 6 ρ11 1 1 -1 1 ζ32 ζ3 -1 ζ32 ζ3 ζ6 ζ3 ζ65 ζ32 1 ζ6 ζ65 1 -1 -1 linear of order 6 ρ12 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 1 1 linear of order 3 ρ13 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ14 2 -2 0 0 -1-√-3 -1+√-3 0 1+√-3 1-√-3 0 0 0 0 2 0 0 -2 0 0 complex lifted from C3×D4 ρ15 2 -2 0 0 -1+√-3 -1-√-3 0 1-√-3 1+√-3 0 0 0 0 2 0 0 -2 0 0 complex lifted from C3×D4 ρ16 6 6 0 0 0 0 6 0 0 0 0 0 0 -1 0 0 -1 -1 -1 orthogonal lifted from F7 ρ17 6 6 0 0 0 0 -6 0 0 0 0 0 0 -1 0 0 -1 1 1 orthogonal lifted from C2×F7 ρ18 6 -6 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 √7 -√7 orthogonal faithful ρ19 6 -6 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 -√7 √7 orthogonal faithful

Permutation representations of C4⋊F7
On 28 points - transitive group 28T23
Generators in S28
```(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)
(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)
(1 15)(2 18 3 21 5 20)(4 17 7 19 6 16)(8 22)(9 25 10 28 12 27)(11 24 14 26 13 23)```

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

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

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

`G:=TransitiveGroup(28,23);`

C4⋊F7 is a maximal subgroup of   C56⋊C6  D56⋊C3  D4⋊F7  Q82F7  D286C6  D4×F7  Q83F7
C4⋊F7 is a maximal quotient of   C56⋊C6  D56⋊C3  C8.F7  C28⋊C12  D14⋊C12

Matrix representation of C4⋊F7 in GL6(𝔽3)

 2 0 2 1 1 0 1 0 2 2 1 2 1 0 1 0 1 0 2 0 1 2 1 0 0 0 2 2 1 0 2 1 2 1 0 0
,
 0 2 0 2 2 2 0 1 0 1 1 0 1 1 0 1 0 0 1 2 0 0 1 0 1 0 2 2 1 0 2 2 0 0 2 0
,
 1 1 0 2 0 0 0 2 1 1 0 0 0 0 0 1 1 0 0 2 0 0 0 1 0 2 0 2 0 0 0 1 0 0 0 0

`G:=sub<GL(6,GF(3))| [2,1,1,2,0,2,0,0,0,0,0,1,2,2,1,1,2,2,1,2,0,2,2,1,1,1,1,1,1,0,0,2,0,0,0,0],[0,0,1,1,1,2,2,1,1,2,0,2,0,0,0,0,2,0,2,1,1,0,2,0,2,1,0,1,1,2,2,0,0,0,0,0],[1,0,0,0,0,0,1,2,0,2,2,1,0,1,0,0,0,0,2,1,1,0,2,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C4⋊F7 in GAP, Magma, Sage, TeX

`C_4\rtimes F_7`
`% in TeX`

`G:=Group("C4:F7");`
`// GroupNames label`

`G:=SmallGroup(168,9);`
`// by ID`

`G=gap.SmallGroup(168,9);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-2,-7,141,66,3604,614]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=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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