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## G = D4⋊F5order 160 = 25·5

### 2nd semidirect product of D4 and F5 acting via F5/D5=C2

Aliases: D42F5, D10.2D4, Dic101C4, Dic5.21D4, C51C4≀C2, (C5×D4)⋊2C4, (C4×F5)⋊1C2, C4.F51C2, C4.2(C2×F5), C20.2(C2×C4), D42D5.2C2, (C4×D5).8C22, C2.7(C22⋊F5), C10.6(C22⋊C4), SmallGroup(160,83)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — D4⋊F5
 Lower central C5 — C10 — C20 — D4⋊F5
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4⋊F5
G = < a,b,c,d | a4=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b, dcd-1=c3 >

Character table of D4⋊F5

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

Smallest permutation representation of D4⋊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 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(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)]])`

D4⋊F5 is a maximal subgroup of
D85F5  D8⋊F5  SD163F5  SD162F5  (C2×D4)⋊6F5  D5⋊C4≀C2  D4⋊F5⋊C2  D124F5  D122F5  Dic10⋊Dic3
D4⋊F5 is a maximal quotient of
D10.1Q16  C10.C4≀C2  Dic101C8  C20.C42  D10.SD16  (C2×D4).F5  Dic5.23D8  D124F5  D122F5  Dic10⋊Dic3

Matrix representation of D4⋊F5 in GL6(𝔽41)

 32 8 0 0 0 0 0 9 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
,
 20 13 0 0 0 0 4 21 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
,
 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
,
 32 40 0 0 0 0 0 40 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))| [32,0,0,0,0,0,8,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,4,0,0,0,0,13,21,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[32,0,0,0,0,0,40,40,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;`

D4⋊F5 in GAP, Magma, Sage, TeX

`D_4\rtimes F_5`
`% in TeX`

`G:=Group("D4:F5");`
`// GroupNames label`

`G:=SmallGroup(160,83);`
`// by ID`

`G=gap.SmallGroup(160,83);`
`# by ID`

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

`G:=Group<a,b,c,d|a^4=b^2=c^5=d^4=1,b*a*b=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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