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## G = D4⋊D5order 80 = 24·5

### The semidirect product of D4 and D5 acting via D5/C5=C2

Aliases: D4⋊D5, C52D8, D202C2, C4.1D10, C10.7D4, C20.1C22, C52C81C2, (C5×D4)⋊1C2, C2.4(C5⋊D4), SmallGroup(80,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊D5
 Chief series C1 — C5 — C10 — C20 — D20 — D4⋊D5
 Lower central C5 — C10 — C20 — D4⋊D5
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4⋊D5
G = < a,b,c,d | a4=b2=c5=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

Character table of D4⋊D5

 class 1 2A 2B 2C 4 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 20A 20B size 1 1 4 20 2 2 2 10 10 2 2 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 0 -2 2 2 0 0 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 0 2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 2 -2 0 2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ8 2 2 2 0 2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 -2 0 2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ10 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ11 2 -2 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 0 -2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ53-ζ52 -ζ54+ζ5 -ζ53+ζ52 ζ54-ζ5 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ13 2 2 0 0 -2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -ζ54+ζ5 -ζ53+ζ52 ζ54-ζ5 ζ53-ζ52 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ14 2 2 0 0 -2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ54-ζ5 ζ53-ζ52 -ζ54+ζ5 -ζ53+ζ52 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ15 2 2 0 0 -2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -ζ53+ζ52 ζ54-ζ5 ζ53-ζ52 -ζ54+ζ5 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 4 -4 0 0 0 -1+√5 -1-√5 0 0 1-√5 1+√5 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ17 4 -4 0 0 0 -1-√5 -1+√5 0 0 1+√5 1-√5 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of D4⋊D5
On 40 points
Generators in S40
```(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 39)(2 40)(3 36)(4 37)(5 38)(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)
(1 5)(2 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)```

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

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

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

D4⋊D5 is a maximal subgroup of
D5×D8  D8⋊D5  D40⋊C2  SD163D5  D4.D10  D4⋊D10  D4.8D10  C15⋊D8  C5⋊D24  D4⋊D15  D4⋊D25  C522D8  C5⋊D40  C527D8
D4⋊D5 is a maximal quotient of
C10.D8  D206C4  C5⋊D16  D8.D5  C5⋊SD32  C5⋊Q32  D4⋊Dic5  C15⋊D8  C5⋊D24  D4⋊D15  D4⋊D25  C522D8  C5⋊D40  C527D8

Matrix representation of D4⋊D5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 39 0 0 1 40
,
 1 0 0 0 0 1 0 0 0 0 0 24 0 0 12 0
,
 34 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 1 34 0 0 0 40 0 0 0 0 1 0 0 0 1 40
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,24,0],[34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,1,1,0,0,0,40] >;`

D4⋊D5 in GAP, Magma, Sage, TeX

`D_4\rtimes D_5`
`% in TeX`

`G:=Group("D4:D5");`
`// GroupNames label`

`G:=SmallGroup(80,15);`
`// by ID`

`G=gap.SmallGroup(80,15);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,61,182,97,42,1604]);`
`// Polycyclic`

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

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