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## G = C5⋊D4order 40 = 23·5

### The semidirect product of C5 and D4 acting via D4/C22=C2

Aliases: C52D4, C22⋊D5, Dic5⋊C2, D102C2, C2.5D10, C10.5C22, (C2×C10)⋊2C2, SmallGroup(40,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5⋊D4
 Chief series C1 — C5 — C10 — D10 — C5⋊D4
 Lower central C5 — C10 — C5⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C5⋊D4
G = < a,b,c | a5=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C5⋊D4

 class 1 2A 2B 2C 4 5A 5B 10A 10B 10C 10D 10E 10F size 1 1 2 10 10 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 -2 0 0 0 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ6 2 2 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 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 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 ρ9 2 2 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 ρ10 2 -2 0 0 0 -1-√5/2 -1+√5/2 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 1-√5/2 1+√5/2 complex faithful ρ11 2 -2 0 0 0 -1-√5/2 -1+√5/2 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 1-√5/2 1+√5/2 complex faithful ρ12 2 -2 0 0 0 -1+√5/2 -1-√5/2 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 1+√5/2 1-√5/2 complex faithful ρ13 2 -2 0 0 0 -1+√5/2 -1-√5/2 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 1+√5/2 1-√5/2 complex faithful

Permutation representations of C5⋊D4
On 20 points - transitive group 20T7
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 14 9 19)(2 13 10 18)(3 12 6 17)(4 11 7 16)(5 15 8 20)
(2 5)(3 4)(6 7)(8 10)(11 17)(12 16)(13 20)(14 19)(15 18)```

`G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,9,19)(2,13,10,18)(3,12,6,17)(4,11,7,16)(5,15,8,20), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,9,19)(2,13,10,18)(3,12,6,17)(4,11,7,16)(5,15,8,20), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18) );`

`G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,14,9,19),(2,13,10,18),(3,12,6,17),(4,11,7,16),(5,15,8,20)], [(2,5),(3,4),(6,7),(8,10),(11,17),(12,16),(13,20),(14,19),(15,18)])`

`G:=TransitiveGroup(20,7);`

On 20 points - transitive group 20T11
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16 9 11)(2 20 10 15)(3 19 6 14)(4 18 7 13)(5 17 8 12)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)```

`G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,9,11)(2,20,10,15)(3,19,6,14)(4,18,7,13)(5,17,8,12), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,9,11)(2,20,10,15)(3,19,6,14)(4,18,7,13)(5,17,8,12), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20) );`

`G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16,9,11),(2,20,10,15),(3,19,6,14),(4,18,7,13),(5,17,8,12)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20)])`

`G:=TransitiveGroup(20,11);`

Matrix representation of C5⋊D4 in GL2(𝔽11) generated by

 5 8 8 2
,
 8 2 6 3
,
 1 1 0 10
`G:=sub<GL(2,GF(11))| [5,8,8,2],[8,6,2,3],[1,0,1,10] >;`

C5⋊D4 in GAP, Magma, Sage, TeX

`C_5\rtimes D_4`
`% in TeX`

`G:=Group("C5:D4");`
`// GroupNames label`

`G:=SmallGroup(40,8);`
`// by ID`

`G=gap.SmallGroup(40,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-5,49,515]);`
`// Polycyclic`

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

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