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## G = Q8⋊F5order 160 = 25·5

### 1st semidirect product of Q8 and F5 acting via F5/D5=C2

Aliases: Q81F5, D5.2Q16, Dic102C4, D10.19D4, Dic5.3D4, D5.3SD16, C5⋊(Q8⋊C4), (C5×Q8)⋊1C4, C4⋊F5.1C2, C4.3(C2×F5), C20.3(C2×C4), D5⋊C8.1C2, (Q8×D5).2C2, (C4×D5).9C22, C2.8(C22⋊F5), C10.7(C22⋊C4), SmallGroup(160,84)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q8⋊F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — Q8⋊F5
 Lower central C5 — C10 — C20 — Q8⋊F5
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q8⋊F5
G = < a,b,c,d | a4=c5=d4=1, b2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c3 >

Character table of Q8⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 10 20A 20B 20C size 1 1 5 5 2 4 10 20 20 20 4 10 10 10 10 4 8 8 8 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 1 -1 -1 -i 1 i 1 i -i -i i 1 -1 -1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -1 -i -1 i 1 -i i i -i 1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -1 i 1 -i 1 -i i i -i 1 -1 -1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 i -1 -i 1 i -i -i i 1 1 1 1 linear of order 4 ρ9 2 2 2 2 -2 0 -2 0 0 0 2 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ10 2 2 -2 -2 -2 0 2 0 0 0 2 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 0 0 0 0 2 -√2 -√2 √2 √2 -2 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 -2 2 0 0 0 0 0 0 2 √2 √2 -√2 -√2 -2 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 0 0 0 0 0 0 2 √-2 -√-2 √-2 -√-2 -2 0 0 0 complex lifted from SD16 ρ14 2 -2 2 -2 0 0 0 0 0 0 2 -√-2 √-2 -√-2 √-2 -2 0 0 0 complex lifted from SD16 ρ15 4 4 0 0 4 4 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ16 4 4 0 0 4 -4 0 0 0 0 -1 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F5 ρ17 4 4 0 0 -4 0 0 0 0 0 -1 0 0 0 0 -1 √5 -√5 1 orthogonal lifted from C22⋊F5 ρ18 4 4 0 0 -4 0 0 0 0 0 -1 0 0 0 0 -1 -√5 √5 1 orthogonal lifted from C22⋊F5 ρ19 8 -8 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8⋊F5
On 40 points
Generators in S40
```(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(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)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 31)(22 33 25 34)(23 35 24 32)(26 36)(27 38 30 39)(28 40 29 37)```

`G:=sub<Sym(40)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31)(22,33,25,34)(23,35,24,32)(26,36)(27,38,30,39)(28,40,29,37)>;`

`G:=Group( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31)(22,33,25,34)(23,35,24,32)(26,36)(27,38,30,39)(28,40,29,37) );`

`G=PermutationGroup([(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,26,6,21),(2,27,7,22),(3,28,8,23),(4,29,9,24),(5,30,10,25),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35)], [(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)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17),(21,31),(22,33,25,34),(23,35,24,32),(26,36),(27,38,30,39),(28,40,29,37)])`

Q8⋊F5 is a maximal subgroup of
SD16×F5  SD16⋊F5  Q16×F5  Dic20⋊C4  (C2×Q8)⋊4F5  C4○D4⋊F5  C4○D20⋊C4  Dic6⋊F5  Dic30⋊C4  Dic102Dic3  D10.S4
Q8⋊F5 is a maximal quotient of
D10.1Q16  Dic101C8  Dic5.D8  D10.18D8  D10.Q16  Dic5.12Q16  Dic5.Q16  Dic6⋊F5  Dic30⋊C4  Dic102Dic3

Matrix representation of Q8⋊F5 in GL6(𝔽41)

 40 39 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 11 0 0 0 0 26 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 32 0 0 0 0 0 9 9 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

`G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,26,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[32,9,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;`

Q8⋊F5 in GAP, Magma, Sage, TeX

`Q_8\rtimes F_5`
`% in TeX`

`G:=Group("Q8:F5");`
`// GroupNames label`

`G:=SmallGroup(160,84);`
`// by ID`

`G=gap.SmallGroup(160,84);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,579,297,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^5=d^4=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;`
`// generators/relations`

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