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## G = D4.F5order 160 = 25·5

### The non-split extension by D4 of F5 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4.F5
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — D4.F5
 Lower central C5 — C10 — D4.F5
 Upper central C1 — C2 — D4

Generators and relations for D4.F5
G = < a,b,c,d | a4=b2=c5=1, d4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 164 in 62 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, D10, C2×C10, C8○D4, C5⋊C8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, D42D5, D4.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, F5, C8○D4, C2×F5, C22×F5, D4.F5

Character table of D4.F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 20 size 1 1 2 2 10 2 5 5 10 10 4 5 5 5 5 10 10 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 1 -1 1 -1 -1 -1 1 -1 1 -i -i i i -i i -i -i i i 1 1 -1 -1 linear of order 4 ρ10 1 1 1 1 -1 1 -1 -1 -1 -1 1 -i -i i i i -i i -i i -i 1 1 1 1 linear of order 4 ρ11 1 1 1 -1 1 -1 -1 -1 1 -1 1 i i -i -i i -i i i -i -i 1 1 -1 -1 linear of order 4 ρ12 1 1 1 1 -1 1 -1 -1 -1 -1 1 i i -i -i -i i -i i -i i 1 1 1 1 linear of order 4 ρ13 1 1 -1 1 1 -1 -1 -1 -1 1 1 i i -i -i i i -i -i i -i 1 -1 1 -1 linear of order 4 ρ14 1 1 -1 -1 -1 1 -1 -1 1 1 1 i i -i -i -i -i i -i i i 1 -1 -1 1 linear of order 4 ρ15 1 1 -1 1 1 -1 -1 -1 -1 1 1 -i -i i i -i -i i i -i i 1 -1 1 -1 linear of order 4 ρ16 1 1 -1 -1 -1 1 -1 -1 1 1 1 -i -i i i i i -i i -i -i 1 -1 -1 1 linear of order 4 ρ17 2 -2 0 0 0 0 2i -2i 0 0 2 2ζ85 2ζ8 2ζ87 2ζ83 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ18 2 -2 0 0 0 0 2i -2i 0 0 2 2ζ8 2ζ85 2ζ83 2ζ87 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ19 2 -2 0 0 0 0 -2i 2i 0 0 2 2ζ83 2ζ87 2ζ8 2ζ85 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ20 2 -2 0 0 0 0 -2i 2i 0 0 2 2ζ87 2ζ83 2ζ85 2ζ8 0 0 0 0 0 0 -2 0 0 0 complex lifted from C8○D4 ρ21 4 4 4 -4 0 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 -4 4 0 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from C2×F5 ρ23 4 4 4 4 0 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ24 4 4 -4 -4 0 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F5 ρ25 8 -8 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D4.F5
On 80 points
Generators in S80
```(1 7 5 3)(2 8 6 4)(9 55 13 51)(10 56 14 52)(11 49 15 53)(12 50 16 54)(17 63 21 59)(18 64 22 60)(19 57 23 61)(20 58 24 62)(25 43 29 47)(26 44 30 48)(27 45 31 41)(28 46 32 42)(33 77 37 73)(34 78 38 74)(35 79 39 75)(36 80 40 76)(65 67 69 71)(66 68 70 72)
(1 71)(2 72)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 17 37 75 57)(2 76 18 58 38)(3 59 77 39 19)(4 40 60 20 78)(5 21 33 79 61)(6 80 22 62 34)(7 63 73 35 23)(8 36 64 24 74)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(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)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)```

`G:=sub<Sym(80)| (1,7,5,3)(2,8,6,4)(9,55,13,51)(10,56,14,52)(11,49,15,53)(12,50,16,54)(17,63,21,59)(18,64,22,60)(19,57,23,61)(20,58,24,62)(25,43,29,47)(26,44,30,48)(27,45,31,41)(28,46,32,42)(33,77,37,73)(34,78,38,74)(35,79,39,75)(36,80,40,76)(65,67,69,71)(66,68,70,72), (1,71)(2,72)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,17,37,75,57)(2,76,18,58,38)(3,59,77,39,19)(4,40,60,20,78)(5,21,33,79,61)(6,80,22,62,34)(7,63,73,35,23)(8,36,64,24,74)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)>;`

`G:=Group( (1,7,5,3)(2,8,6,4)(9,55,13,51)(10,56,14,52)(11,49,15,53)(12,50,16,54)(17,63,21,59)(18,64,22,60)(19,57,23,61)(20,58,24,62)(25,43,29,47)(26,44,30,48)(27,45,31,41)(28,46,32,42)(33,77,37,73)(34,78,38,74)(35,79,39,75)(36,80,40,76)(65,67,69,71)(66,68,70,72), (1,71)(2,72)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,17,37,75,57)(2,76,18,58,38)(3,59,77,39,19)(4,40,60,20,78)(5,21,33,79,61)(6,80,22,62,34)(7,63,73,35,23)(8,36,64,24,74)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80) );`

`G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,55,13,51),(10,56,14,52),(11,49,15,53),(12,50,16,54),(17,63,21,59),(18,64,22,60),(19,57,23,61),(20,58,24,62),(25,43,29,47),(26,44,30,48),(27,45,31,41),(28,46,32,42),(33,77,37,73),(34,78,38,74),(35,79,39,75),(36,80,40,76),(65,67,69,71),(66,68,70,72)], [(1,71),(2,72),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,53),(18,54),(19,55),(20,56),(21,49),(22,50),(23,51),(24,52),(25,79),(26,80),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,17,37,75,57),(2,76,18,58,38),(3,59,77,39,19),(4,40,60,20,78),(5,21,33,79,61),(6,80,22,62,34),(7,63,73,35,23),(8,36,64,24,74),(9,71,53,45,29),(10,46,72,30,54),(11,31,47,55,65),(12,56,32,66,48),(13,67,49,41,25),(14,42,68,26,50),(15,27,43,51,69),(16,52,28,70,44)], [(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),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)]])`

Matrix representation of D4.F5 in GL6(𝔽41)

 9 0 0 0 0 0 28 32 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
,
 29 37 0 0 0 0 5 12 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
,
 14 0 0 0 0 0 0 14 0 0 0 0 0 0 32 0 18 18 0 0 18 18 0 32 0 0 23 14 23 0 0 0 9 27 27 9

`G:=sub<GL(6,GF(41))| [9,28,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,5,0,0,0,0,37,12,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[14,0,0,0,0,0,0,14,0,0,0,0,0,0,32,18,23,9,0,0,0,18,14,27,0,0,18,0,23,27,0,0,18,32,0,9] >;`

D4.F5 in GAP, Magma, Sage, TeX

`D_4.F_5`
`% in TeX`

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

`G:=SmallGroup(160,206);`
`// by ID`

`G=gap.SmallGroup(160,206);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,69,2309,599]);`
`// Polycyclic`

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

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