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## G = D4.F7order 336 = 24·3·7

### The non-split extension by D4 of F7 acting via F7/C7⋊C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4.F7
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C4.F7 — D4.F7
 Lower central C7 — C14 — C28 — D4.F7
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.F7
G = < a,b,c,d | a4=b2=c7=1, d6=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c5 >

Character table of D4.F7

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 7 8A 8B 12A 12B 12C 12D 14A 14B 14C 24A 24B 24C 24D 28 size 1 1 4 7 7 2 28 7 7 28 28 6 14 14 14 14 28 28 6 12 12 14 14 14 14 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 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 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 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 linear of order 2 ρ5 1 1 1 ζ32 ζ3 1 -1 ζ3 ζ32 ζ32 ζ3 1 -1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 ζ6 ζ6 ζ65 ζ65 1 linear of order 6 ρ6 1 1 -1 ζ32 ζ3 1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 ζ32 ζ3 ζ65 ζ6 1 -1 -1 ζ32 ζ32 ζ3 ζ3 1 linear of order 6 ρ7 1 1 -1 ζ3 ζ32 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 ζ3 ζ32 ζ6 ζ65 1 -1 -1 ζ3 ζ3 ζ32 ζ32 1 linear of order 6 ρ8 1 1 1 ζ3 ζ32 1 -1 ζ32 ζ3 ζ3 ζ32 1 -1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 ζ65 ζ65 ζ6 ζ6 1 linear of order 6 ρ9 1 1 -1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 1 -1 -1 ζ32 ζ3 ζ3 ζ32 1 -1 -1 ζ6 ζ6 ζ65 ζ65 1 linear of order 6 ρ10 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ32 ζ32 ζ3 ζ3 1 linear of order 3 ρ11 1 1 -1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 1 -1 -1 ζ3 ζ32 ζ32 ζ3 1 -1 -1 ζ65 ζ65 ζ6 ζ6 1 linear of order 6 ρ12 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ3 ζ3 ζ32 ζ32 1 linear of order 3 ρ13 2 2 0 2 2 -2 0 2 2 0 0 2 0 0 -2 -2 0 0 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ14 2 2 0 -1+√-3 -1-√-3 -2 0 -1-√-3 -1+√-3 0 0 2 0 0 1-√-3 1+√-3 0 0 2 0 0 0 0 0 0 -2 complex lifted from C3×D4 ρ15 2 2 0 -1-√-3 -1+√-3 -2 0 -1+√-3 -1-√-3 0 0 2 0 0 1+√-3 1-√-3 0 0 2 0 0 0 0 0 0 -2 complex lifted from C3×D4 ρ16 2 -2 0 2 2 0 0 -2 -2 0 0 2 √-2 -√-2 0 0 0 0 -2 0 0 -√-2 √-2 -√-2 √-2 0 complex lifted from SD16 ρ17 2 -2 0 2 2 0 0 -2 -2 0 0 2 -√-2 √-2 0 0 0 0 -2 0 0 √-2 -√-2 √-2 -√-2 0 complex lifted from SD16 ρ18 2 -2 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 2 √-2 -√-2 0 0 0 0 -2 0 0 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 0 complex lifted from C3×SD16 ρ19 2 -2 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 2 -√-2 √-2 0 0 0 0 -2 0 0 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 0 complex lifted from C3×SD16 ρ20 2 -2 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 2 -√-2 √-2 0 0 0 0 -2 0 0 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 0 complex lifted from C3×SD16 ρ21 2 -2 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 2 √-2 -√-2 0 0 0 0 -2 0 0 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 0 complex lifted from C3×SD16 ρ22 6 6 6 0 0 6 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from F7 ρ23 6 6 -6 0 0 6 0 0 0 0 0 -1 0 0 0 0 0 0 -1 1 1 0 0 0 0 -1 orthogonal lifted from C2×F7 ρ24 6 6 0 0 0 -6 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -√-7 √-7 0 0 0 0 1 complex lifted from Dic7⋊C6 ρ25 6 6 0 0 0 -6 0 0 0 0 0 -1 0 0 0 0 0 0 -1 √-7 -√-7 0 0 0 0 1 complex lifted from Dic7⋊C6 ρ26 12 -12 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D4.F7
On 56 points
Generators in S56
```(1 5 3 7)(2 8 4 6)(9 55 15 49)(10 50 16 56)(11 45 17 51)(12 52 18 46)(13 47 19 53)(14 54 20 48)(21 36 27 42)(22 43 28 37)(23 38 29 44)(24 33 30 39)(25 40 31 34)(26 35 32 41)
(1 7)(2 4)(3 5)(9 55)(11 45)(13 47)(15 49)(17 51)(19 53)(21 27)(22 37)(23 29)(24 39)(25 31)(26 41)(28 43)(30 33)(32 35)(46 52)(48 54)(50 56)
(1 49 45 28 53 32 24)(2 21 29 50 25 54 46)(3 55 51 22 47 26 30)(4 27 23 56 31 48 52)(5 9 17 37 13 41 33)(6 42 38 10 34 14 18)(7 15 11 43 19 35 39)(8 36 44 16 40 20 12)
(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)| (1,5,3,7)(2,8,4,6)(9,55,15,49)(10,50,16,56)(11,45,17,51)(12,52,18,46)(13,47,19,53)(14,54,20,48)(21,36,27,42)(22,43,28,37)(23,38,29,44)(24,33,30,39)(25,40,31,34)(26,35,32,41), (1,7)(2,4)(3,5)(9,55)(11,45)(13,47)(15,49)(17,51)(19,53)(21,27)(22,37)(23,29)(24,39)(25,31)(26,41)(28,43)(30,33)(32,35)(46,52)(48,54)(50,56), (1,49,45,28,53,32,24)(2,21,29,50,25,54,46)(3,55,51,22,47,26,30)(4,27,23,56,31,48,52)(5,9,17,37,13,41,33)(6,42,38,10,34,14,18)(7,15,11,43,19,35,39)(8,36,44,16,40,20,12), (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( (1,5,3,7)(2,8,4,6)(9,55,15,49)(10,50,16,56)(11,45,17,51)(12,52,18,46)(13,47,19,53)(14,54,20,48)(21,36,27,42)(22,43,28,37)(23,38,29,44)(24,33,30,39)(25,40,31,34)(26,35,32,41), (1,7)(2,4)(3,5)(9,55)(11,45)(13,47)(15,49)(17,51)(19,53)(21,27)(22,37)(23,29)(24,39)(25,31)(26,41)(28,43)(30,33)(32,35)(46,52)(48,54)(50,56), (1,49,45,28,53,32,24)(2,21,29,50,25,54,46)(3,55,51,22,47,26,30)(4,27,23,56,31,48,52)(5,9,17,37,13,41,33)(6,42,38,10,34,14,18)(7,15,11,43,19,35,39)(8,36,44,16,40,20,12), (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([[(1,5,3,7),(2,8,4,6),(9,55,15,49),(10,50,16,56),(11,45,17,51),(12,52,18,46),(13,47,19,53),(14,54,20,48),(21,36,27,42),(22,43,28,37),(23,38,29,44),(24,33,30,39),(25,40,31,34),(26,35,32,41)], [(1,7),(2,4),(3,5),(9,55),(11,45),(13,47),(15,49),(17,51),(19,53),(21,27),(22,37),(23,29),(24,39),(25,31),(26,41),(28,43),(30,33),(32,35),(46,52),(48,54),(50,56)], [(1,49,45,28,53,32,24),(2,21,29,50,25,54,46),(3,55,51,22,47,26,30),(4,27,23,56,31,48,52),(5,9,17,37,13,41,33),(6,42,38,10,34,14,18),(7,15,11,43,19,35,39),(8,36,44,16,40,20,12)], [(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)]])`

Matrix representation of D4.F7 in GL8(𝔽337)

 0 1 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 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 0 336 124 1 0 0 0 0 0 0 124 1 0 0 0 0 0 336 125 1 0 0 0 0 0 0 0 0 336 1 0 0 0 0 0 0 336 0 1 0 0 0 0 0 124 1 213
,
 262 75 0 0 0 0 0 0 75 75 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 125 1 213 0 0 0 0 0 1 213 336 0 0 0 1 0 0 0 0 0 0 125 1 213 0 0 0 0 0 1 213 336 0 0 0

`G:=sub<GL(8,GF(337))| [0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,0,336,0,0,0,0,0,124,124,125,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,336,336,124,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,213],[262,75,0,0,0,0,0,0,75,75,0,0,0,0,0,0,0,0,0,0,0,0,125,1,0,0,0,0,0,1,1,213,0,0,0,0,0,0,213,336,0,0,0,125,1,0,0,0,0,0,1,1,213,0,0,0,0,0,0,213,336,0,0,0] >;`

D4.F7 in GAP, Magma, Sage, TeX

`D_4.F_7`
`% in TeX`

`G:=Group("D4.F7");`
`// GroupNames label`

`G:=SmallGroup(336,19);`
`// by ID`

`G=gap.SmallGroup(336,19);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,867,441,69,10373,1745]);`
`// Polycyclic`

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

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