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## G = Q8⋊2F5order 160 = 25·5

### 2nd semidirect product of Q8 and F5 acting via F5/D5=C2

Aliases: Q82F5, D202C4, D10.3D4, Dic5.22D4, C52C4≀C2, (C4×F5)⋊2C2, (C5×Q8)⋊2C4, C4.F52C2, C4.4(C2×F5), C20.4(C2×C4), Q82D5.2C2, C2.9(C22⋊F5), C10.8(C22⋊C4), (C4×D5).10C22, SmallGroup(160,85)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q8⋊2F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — Q8⋊2F5
 Lower central C5 — C10 — C20 — Q8⋊2F5
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q82F5
G = < a,b,c,d | a4=c5=d4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b, dcd-1=c3 >

10C2
20C2
2C4
5C22
5C4
10C4
10C22
10C4
2D5
4D5
5D4
10D4
10C2×C4
10C2×C4
10C8
2C20
2F5
2D10
2F5
5C42
2D20

Character table of Q82F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 10 20A 20B 20C size 1 1 10 20 2 4 5 5 10 10 10 10 4 20 20 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 -1 i -i i -i 1 i -i 1 1 -1 -1 linear of order 4 ρ6 1 1 -1 1 1 -1 -1 -1 -i i -i i 1 -i i 1 1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 1 -1 -1 i -i i -i 1 -i i 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 -1 -i i -i i 1 i -i 1 1 1 1 linear of order 4 ρ9 2 2 -2 0 -2 0 2 2 0 0 0 0 2 0 0 2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 -2 0 -2 -2 0 0 0 0 2 0 0 2 -2 0 0 orthogonal lifted from D4 ρ11 2 -2 0 0 0 0 -2i 2i -1+i -1-i 1-i 1+i 2 0 0 -2 0 0 0 complex lifted from C4≀C2 ρ12 2 -2 0 0 0 0 2i -2i -1-i -1+i 1+i 1-i 2 0 0 -2 0 0 0 complex lifted from C4≀C2 ρ13 2 -2 0 0 0 0 2i -2i 1+i 1-i -1-i -1+i 2 0 0 -2 0 0 0 complex lifted from C4≀C2 ρ14 2 -2 0 0 0 0 -2i 2i 1-i 1+i -1+i -1-i 2 0 0 -2 0 0 0 complex lifted from C4≀C2 ρ15 4 4 0 0 4 -4 0 0 0 0 0 0 -1 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ16 4 4 0 0 4 4 0 0 0 0 0 0 -1 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 4 0 0 -4 0 0 0 0 0 0 0 -1 0 0 -1 1 -√5 √5 orthogonal lifted from C22⋊F5 ρ18 4 4 0 0 -4 0 0 0 0 0 0 0 -1 0 0 -1 1 √5 -√5 orthogonal lifted from C22⋊F5 ρ19 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 0 0 orthogonal faithful

Smallest permutation representation of Q82F5
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)(12 13 15 14)(17 18 20 19)(21 36 26 31)(22 38 30 34)(23 40 29 32)(24 37 28 35)(25 39 27 33)```

`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)(12,13,15,14)(17,18,20,19)(21,36,26,31)(22,38,30,34)(23,40,29,32)(24,37,28,35)(25,39,27,33)>;`

`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)(12,13,15,14)(17,18,20,19)(21,36,26,31)(22,38,30,34)(23,40,29,32)(24,37,28,35)(25,39,27,33) );`

`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),(12,13,15,14),(17,18,20,19),(21,36,26,31),(22,38,30,34),(23,40,29,32),(24,37,28,35),(25,39,27,33)]])`

Q82F5 is a maximal subgroup of
SD163F5  SD162F5  Q165F5  Q16⋊F5  (C2×Q8)⋊6F5  D5⋊C4≀C2  D4⋊F5⋊C2  D602C4  D605C4  D202Dic3  C5⋊U2(𝔽3)
Q82F5 is a maximal quotient of
D10.1D8  C10.C4≀C2  D20⋊C8  C20.C42  D10.Q16  (C2×Q8).F5  Dic5.12Q16  D602C4  D605C4  D202Dic3

Matrix representation of Q82F5 in GL6(𝔽41)

 9 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 22 0 3 3 0 0 38 19 38 0 0 0 0 38 19 38 0 0 3 3 0 22
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;`

Q82F5 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2F_5`
`% in TeX`

`G:=Group("Q8:2F5");`
`// GroupNames label`

`G:=SmallGroup(160,85);`
`// by ID`

`G=gap.SmallGroup(160,85);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,86,579,297,69,2309,1169]);`
`// Polycyclic`

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

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