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## G = Q8⋊3F7order 336 = 24·3·7

### The semidirect product of Q8 and F7 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Q8⋊3F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4×F7 — Q8⋊3F7
 Lower central C7 — C14 — Q8⋊3F7
 Upper central C1 — C2 — Q8

Generators and relations for Q83F7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 332 in 80 conjugacy classes, 40 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2×C4, D4, Q8, C12, C2×C6, D7, C14, C4○D4, C7⋊C3, C2×C12, C3×D4, C3×Q8, Dic7, C28, D14, F7, C2×C7⋊C3, C3×C4○D4, C4×D7, D28, C7×Q8, C7⋊C12, C4×C7⋊C3, C2×F7, Q82D7, C4×F7, C4⋊F7, Q8×C7⋊C3, Q83F7
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C22×C6, F7, C3×C4○D4, C2×F7, C22×F7, Q83F7

Smallest permutation representation of Q83F7
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 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
(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 43)(2 46 3 49 5 48)(4 45 7 47 6 44)(8 50)(9 53 10 56 12 55)(11 52 14 54 13 51)(15 36)(16 39 17 42 19 41)(18 38 21 40 20 37)(22 29)(23 32 24 35 26 34)(25 31 28 33 27 30)```

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 7 12A 12B 12C 12D 12E ··· 12J 14 28A 28B 28C order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 7 12 12 12 12 12 ··· 12 14 28 28 28 size 1 1 14 14 14 7 7 2 2 2 7 7 7 7 14 ··· 14 6 7 7 7 7 14 ··· 14 6 12 12 12

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 6 6 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 Q8⋊3F7 C4○D4 C3×C4○D4 F7 C2×F7 kernel Q8⋊3F7 C4×F7 C4⋊F7 Q8×C7⋊C3 Q8⋊2D7 C4×D7 D28 C7×Q8 C1 C7⋊C3 C7 Q8 C4 # reps 1 3 3 1 2 6 6 2 1 2 4 1 3

Matrix representation of Q83F7 in GL8(𝔽337)

 189 336 0 0 0 0 0 0 0 148 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 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
,
 189 0 0 0 0 0 0 0 335 148 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 336 336 336 336 336 0 0 1 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 0 0 0 0 0 0 0 0 1 0
,
 209 0 0 0 0 0 0 0 144 128 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 336 0

`G:=sub<GL(8,GF(337))| [189,0,0,0,0,0,0,0,336,148,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[189,335,0,0,0,0,0,0,0,148,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[209,144,0,0,0,0,0,0,0,128,0,0,0,0,0,0,0,0,336,0,0,0,1,0,0,0,0,0,0,336,1,0,0,0,0,0,0,0,1,0,0,0,0,0,336,0,1,0,0,0,0,0,0,0,1,336,0,0,0,336,0,0,1,0] >;`

Q83F7 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3F_7`
`% in TeX`

`G:=Group("Q8:3F7");`
`// GroupNames label`

`G:=SmallGroup(336,128);`
`// by ID`

`G=gap.SmallGroup(336,128);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,506,260,122,10373,887]);`
`// Polycyclic`

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

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