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## G = D4⋊F7order 336 = 24·3·7

### The semidirect product of D4 and F7 acting via F7/C7⋊C3=C2

Aliases: D4⋊F7, D281C6, D4⋊D7⋊C3, C7⋊C32D8, C7⋊C81C6, C72(C3×D8), C7⋊C241C2, C4⋊F71C2, (C7×D4)⋊1C6, C4.1(C2×F7), C28.1(C2×C6), C14.7(C3×D4), C2.4(Dic7⋊C6), (D4×C7⋊C3)⋊1C2, (C2×C7⋊C3).7D4, (C4×C7⋊C3).1C22, SmallGroup(336,18)

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=d6=1, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c5 >

Character table of D4⋊F7

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7 8A 8B 12A 12B 14A 14B 14C 24A 24B 24C 24D 28 size 1 1 4 28 7 7 2 7 7 28 28 28 28 6 14 14 14 14 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 1 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ6 ζ65 ζ3 1 -1 -1 ζ32 ζ3 1 -1 -1 ζ65 ζ65 ζ6 ζ6 1 linear of order 6 ρ6 1 1 -1 1 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ65 ζ6 ζ32 1 -1 -1 ζ3 ζ32 1 -1 -1 ζ6 ζ6 ζ65 ζ65 1 linear of order 6 ρ7 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ3 ζ32 ζ32 1 linear of order 3 ρ8 1 1 1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ65 ζ3 ζ32 ζ6 1 -1 -1 ζ3 ζ32 1 1 1 ζ6 ζ6 ζ65 ζ65 1 linear of order 6 ρ9 1 1 1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ6 ζ32 ζ3 ζ65 1 -1 -1 ζ32 ζ3 1 1 1 ζ65 ζ65 ζ6 ζ6 1 linear of order 6 ρ10 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ32 ζ3 ζ3 1 linear of order 3 ρ11 1 1 -1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 1 1 1 ζ32 ζ3 1 -1 -1 ζ3 ζ3 ζ32 ζ32 1 linear of order 6 ρ12 1 1 -1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 1 1 1 ζ3 ζ32 1 -1 -1 ζ32 ζ32 ζ3 ζ3 1 linear of order 6 ρ13 2 2 0 0 2 2 -2 2 2 0 0 0 0 2 0 0 -2 -2 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ14 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 2 -√2 √2 0 0 -2 0 0 √2 -√2 √2 -√2 0 orthogonal lifted from D8 ρ15 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 2 √2 -√2 0 0 -2 0 0 -√2 √2 -√2 √2 0 orthogonal lifted from D8 ρ16 2 2 0 0 -1-√-3 -1+√-3 -2 -1+√-3 -1-√-3 0 0 0 0 2 0 0 1+√-3 1-√-3 2 0 0 0 0 0 0 -2 complex lifted from C3×D4 ρ17 2 2 0 0 -1+√-3 -1-√-3 -2 -1-√-3 -1+√-3 0 0 0 0 2 0 0 1-√-3 1+√-3 2 0 0 0 0 0 0 -2 complex lifted from C3×D4 ρ18 2 -2 0 0 -1-√-3 -1+√-3 0 1-√-3 1+√-3 0 0 0 0 2 √2 -√2 0 0 -2 0 0 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 0 complex lifted from C3×D8 ρ19 2 -2 0 0 -1+√-3 -1-√-3 0 1+√-3 1-√-3 0 0 0 0 2 √2 -√2 0 0 -2 0 0 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 0 complex lifted from C3×D8 ρ20 2 -2 0 0 -1+√-3 -1-√-3 0 1+√-3 1-√-3 0 0 0 0 2 -√2 √2 0 0 -2 0 0 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 0 complex lifted from C3×D8 ρ21 2 -2 0 0 -1-√-3 -1+√-3 0 1-√-3 1+√-3 0 0 0 0 2 -√2 √2 0 0 -2 0 0 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 0 complex lifted from C3×D8 ρ22 6 6 -6 0 0 0 6 0 0 0 0 0 0 -1 0 0 0 0 -1 1 1 0 0 0 0 -1 orthogonal lifted from C2×F7 ρ23 6 6 6 0 0 0 6 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from F7 ρ24 6 6 0 0 0 0 -6 0 0 0 0 0 0 -1 0 0 0 0 -1 √-7 -√-7 0 0 0 0 1 complex lifted from Dic7⋊C6 ρ25 6 6 0 0 0 0 -6 0 0 0 0 0 0 -1 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 0 0 -2 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal faithful

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

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

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

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

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 64 325 261 1 0 0 0 0 64 325 261 0 1 0 0 0 64 325 261 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 212 1 0 0 0 0 0 0 212 1 0 0 0 0 0 336 213 1 0 0 0 0 0 26 217 311 0 1 0 0 0 26 217 311 0 0 1 0 0 243 71 285 1 125 124
,
 21 316 0 0 0 0 0 0 316 316 0 0 0 0 0 0 0 0 0 0 0 336 0 1 0 0 165 285 120 123 336 336 0 0 120 146 26 211 213 1 0 0 79 259 180 26 0 0 0 0 79 260 180 26 0 0 0 0 292 260 305 26 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],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,64,64,64,0,0,0,336,0,325,325,325,0,0,0,0,336,261,261,261,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,0,336,26,26,243,0,0,212,212,213,217,217,71,0,0,1,1,1,311,311,285,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,125,0,0,0,0,0,0,1,124],[21,316,0,0,0,0,0,0,316,316,0,0,0,0,0,0,0,0,0,165,120,79,79,292,0,0,0,285,146,259,260,260,0,0,0,120,26,180,180,305,0,0,336,123,211,26,26,26,0,0,0,336,213,0,0,0,0,0,1,336,1,0,0,0] >;`

D4⋊F7 in GAP, Magma, Sage, TeX

`D_4\rtimes F_7`
`% in TeX`

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

`G:=SmallGroup(336,18);`
`// by ID`

`G=gap.SmallGroup(336,18);`
`# by ID`

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

`G:=Group<a,b,c,d|a^4=b^2=c^7=d^6=1,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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