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## G = C4.F7order 168 = 23·3·7

### The non-split extension by C4 of F7 acting via F7/C7⋊C3=C2

Aliases: C4.F7, Dic14⋊C3, C28.1C6, Dic7.C6, C7⋊C3⋊Q8, C7⋊(C3×Q8), C7⋊C12.C2, C2.3(C2×F7), C14.1(C2×C6), (C4×C7⋊C3).1C2, (C2×C7⋊C3).1C22, SmallGroup(168,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4.F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C7⋊C12 — 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=1, c6=a2, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C4.F7

 class 1 2 3A 3B 4A 4B 4C 6A 6B 7 12A 12B 12C 12D 12E 12F 14 28A 28B size 1 1 7 7 2 14 14 7 7 6 14 14 14 14 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 ζ32 ζ3 -1 1 -1 ζ32 ζ3 1 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 1 -1 -1 linear of order 6 ρ6 1 1 ζ3 ζ32 -1 1 -1 ζ3 ζ32 1 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 1 -1 -1 linear of order 6 ρ7 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 linear of order 3 ρ8 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 linear of order 3 ρ9 1 1 ζ3 ζ32 -1 -1 1 ζ3 ζ32 1 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 1 -1 -1 linear of order 6 ρ10 1 1 ζ32 ζ3 -1 -1 1 ζ32 ζ3 1 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 1 -1 -1 linear of order 6 ρ11 1 1 ζ32 ζ3 1 -1 -1 ζ32 ζ3 1 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 1 linear of order 6 ρ12 1 1 ζ3 ζ32 1 -1 -1 ζ3 ζ32 1 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 1 linear of order 6 ρ13 2 -2 2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -1+√-3 -1-√-3 0 0 0 1-√-3 1+√-3 2 0 0 0 0 0 0 -2 0 0 complex lifted from C3×Q8 ρ15 2 -2 -1-√-3 -1+√-3 0 0 0 1+√-3 1-√-3 2 0 0 0 0 0 0 -2 0 0 complex lifted from C3×Q8 ρ16 6 6 0 0 -6 0 0 0 0 -1 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C2×F7 ρ17 6 6 0 0 6 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ18 6 -6 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -√7 √7 symplectic faithful, Schur index 2 ρ19 6 -6 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 √7 -√7 symplectic faithful, Schur index 2

Smallest permutation representation of C4.F7
On 56 points
Generators in S56
```(1 7 3 5)(2 6 4 8)(9 35 15 41)(10 42 16 36)(11 37 17 43)(12 44 18 38)(13 39 19 33)(14 34 20 40)(21 51 27 45)(22 46 28 52)(23 53 29 47)(24 48 30 54)(25 55 31 49)(26 50 32 56)
(1 22 30 41 26 33 37)(2 34 42 23 38 27 31)(3 28 24 35 32 39 43)(4 40 36 29 44 21 25)(5 52 48 15 56 19 11)(6 20 16 53 12 45 49)(7 46 54 9 50 13 17)(8 14 10 47 18 51 55)
(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 36 37 38 39 40 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55 56)```

`G:=sub<Sym(56)| (1,7,3,5)(2,6,4,8)(9,35,15,41)(10,42,16,36)(11,37,17,43)(12,44,18,38)(13,39,19,33)(14,34,20,40)(21,51,27,45)(22,46,28,52)(23,53,29,47)(24,48,30,54)(25,55,31,49)(26,50,32,56), (1,22,30,41,26,33,37)(2,34,42,23,38,27,31)(3,28,24,35,32,39,43)(4,40,36,29,44,21,25)(5,52,48,15,56,19,11)(6,20,16,53,12,45,49)(7,46,54,9,50,13,17)(8,14,10,47,18,51,55), (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)(33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56)>;`

`G:=Group( (1,7,3,5)(2,6,4,8)(9,35,15,41)(10,42,16,36)(11,37,17,43)(12,44,18,38)(13,39,19,33)(14,34,20,40)(21,51,27,45)(22,46,28,52)(23,53,29,47)(24,48,30,54)(25,55,31,49)(26,50,32,56), (1,22,30,41,26,33,37)(2,34,42,23,38,27,31)(3,28,24,35,32,39,43)(4,40,36,29,44,21,25)(5,52,48,15,56,19,11)(6,20,16,53,12,45,49)(7,46,54,9,50,13,17)(8,14,10,47,18,51,55), (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)(33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56) );`

`G=PermutationGroup([(1,7,3,5),(2,6,4,8),(9,35,15,41),(10,42,16,36),(11,37,17,43),(12,44,18,38),(13,39,19,33),(14,34,20,40),(21,51,27,45),(22,46,28,52),(23,53,29,47),(24,48,30,54),(25,55,31,49),(26,50,32,56)], [(1,22,30,41,26,33,37),(2,34,42,23,38,27,31),(3,28,24,35,32,39,43),(4,40,36,29,44,21,25),(5,52,48,15,56,19,11),(6,20,16,53,12,45,49),(7,46,54,9,50,13,17),(8,14,10,47,18,51,55)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44),(45,46,47,48,49,50,51,52,53,54,55,56)])`

C4.F7 is a maximal subgroup of   C56⋊C6  C8.F7  D4.F7  Q8.2F7  D286C6  D42F7  Q8×F7
C4.F7 is a maximal quotient of   Dic7⋊C12  C28⋊C12

Matrix representation of C4.F7 in GL6(𝔽3)

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

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

C4.F7 in GAP, Magma, Sage, TeX

`C_4.F_7`
`% in TeX`

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

`G:=SmallGroup(168,7);`
`// by ID`

`G=gap.SmallGroup(168,7);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-2,-7,60,141,66,3604,614]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^7=1,c^6=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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