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## G = D5≀C2order 200 = 23·52

### Wreath product of D5 by C2

Aliases: D5C2, C52⋊D4, D52⋊C2, C52⋊C42C2, C5⋊D5.2C22, SmallGroup(200,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — D5≀C2
 Chief series C1 — C52 — C5⋊D5 — D52 — D5≀C2
 Lower central C52 — C5⋊D5 — D5≀C2
 Upper central C1

Generators and relations for D5≀C2
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, cac-1=dbd=a2, dad=cbc-1=b3, dcd=c-1 >

10C2
10C2
25C2
2C5
2C5
2C5
25C22
25C4
25C22
2D5
2D5
10D5
10D5
10C10
10C10
10D5
25D4
10D10
10D10
10F5

Character table of D5≀C2

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E 10A 10B 10C 10D size 1 10 10 25 50 4 4 4 4 8 20 20 20 20 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 0 0 -2 0 2 2 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ6 4 0 -2 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 1+√5/2 1-√5/2 0 orthogonal faithful ρ7 4 0 -2 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 1-√5/2 1+√5/2 0 orthogonal faithful ρ8 4 2 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 -1+√5/2 0 0 -1-√5/2 orthogonal faithful ρ9 4 2 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 -1-√5/2 0 0 -1+√5/2 orthogonal faithful ρ10 4 0 2 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 -1+√5/2 -1-√5/2 0 orthogonal faithful ρ11 4 0 2 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 -1-√5/2 -1+√5/2 0 orthogonal faithful ρ12 4 -2 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 1+√5/2 0 0 1-√5/2 orthogonal faithful ρ13 4 -2 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 1-√5/2 0 0 1+√5/2 orthogonal faithful ρ14 8 0 0 0 0 -2 -2 -2 -2 3 0 0 0 0 orthogonal faithful

Permutation representations of D5≀C2
On 10 points - transitive group 10T19
Generators in S10
```(6 7 8 9 10)
(1 5 4 3 2)
(2 3 5 4)(7 9 10 8)
(1 6)(2 9)(3 7)(4 10)(5 8)```

`G:=sub<Sym(10)| (6,7,8,9,10), (1,5,4,3,2), (2,3,5,4)(7,9,10,8), (1,6)(2,9)(3,7)(4,10)(5,8)>;`

`G:=Group( (6,7,8,9,10), (1,5,4,3,2), (2,3,5,4)(7,9,10,8), (1,6)(2,9)(3,7)(4,10)(5,8) );`

`G=PermutationGroup([[(6,7,8,9,10)], [(1,5,4,3,2)], [(2,3,5,4),(7,9,10,8)], [(1,6),(2,9),(3,7),(4,10),(5,8)]])`

`G:=TransitiveGroup(10,19);`

On 10 points - transitive group 10T21
Generators in S10
```(1 2 3 4 5)(6 7 8 9 10)
(1 2 3 4 5)(6 10 9 8 7)
(1 8)(2 6 5 10)(3 9 4 7)
(1 8)(2 10)(3 7)(4 9)(5 6)```

`G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,2,3,4,5)(6,10,9,8,7), (1,8)(2,6,5,10)(3,9,4,7), (1,8)(2,10)(3,7)(4,9)(5,6)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,2,3,4,5)(6,10,9,8,7), (1,8)(2,6,5,10)(3,9,4,7), (1,8)(2,10)(3,7)(4,9)(5,6) );`

`G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,2,3,4,5),(6,10,9,8,7)], [(1,8),(2,6,5,10),(3,9,4,7)], [(1,8),(2,10),(3,7),(4,9),(5,6)]])`

`G:=TransitiveGroup(10,21);`

On 20 points - transitive group 20T48
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 2 3 4 5)(6 7 8 9 10)(11 15 14 13 12)(16 20 19 18 17)
(1 17 6 13)(2 20 10 15)(3 18 9 12)(4 16 8 14)(5 19 7 11)
(1 13)(2 15)(3 12)(4 14)(5 11)(6 17)(7 19)(8 16)(9 18)(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), (1,2,3,4,5)(6,7,8,9,10)(11,15,14,13,12)(16,20,19,18,17), (1,17,6,13)(2,20,10,15)(3,18,9,12)(4,16,8,14)(5,19,7,11), (1,13)(2,15)(3,12)(4,14)(5,11)(6,17)(7,19)(8,16)(9,18)(10,20)>;`

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

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

`G:=TransitiveGroup(20,48);`

On 20 points - transitive group 20T50
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 9 7 10 8)(11 13 15 12 14)(16 19 17 20 18)
(1 16)(2 19 5 18)(3 17 4 20)(6 14 7 12)(8 15 10 11)(9 13)
(1 13)(2 14)(3 15)(4 11)(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), (1,3,5,2,4)(6,9,7,10,8)(11,13,15,12,14)(16,19,17,20,18), (1,16)(2,19,5,18)(3,17,4,20)(6,14,7,12)(8,15,10,11)(9,13), (1,13)(2,14)(3,15)(4,11)(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), (1,3,5,2,4)(6,9,7,10,8)(11,13,15,12,14)(16,19,17,20,18), (1,16)(2,19,5,18)(3,17,4,20)(6,14,7,12)(8,15,10,11)(9,13), (1,13)(2,14)(3,15)(4,11)(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)], [(1,3,5,2,4),(6,9,7,10,8),(11,13,15,12,14),(16,19,17,20,18)], [(1,16),(2,19,5,18),(3,17,4,20),(6,14,7,12),(8,15,10,11),(9,13)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20)]])`

`G:=TransitiveGroup(20,50);`

On 20 points - transitive group 20T55
Generators in S20
```(11 12 13 14 15)(16 17 18 19 20)
(1 5 4 3 2)(6 9 7 10 8)
(1 8)(2 7 5 9)(3 6 4 10)(11 17 13 18)(12 20)(14 16 15 19)
(1 20)(2 18)(3 16)(4 19)(5 17)(6 14)(7 13)(8 12)(9 11)(10 15)```

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

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

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

`G:=TransitiveGroup(20,55);`

On 20 points - transitive group 20T57
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 17 9 13)(2 20 8 15)(3 18 7 12)(4 16 6 14)(5 19 10 11)
(1 13)(2 11)(3 14)(4 12)(5 15)(6 18)(7 16)(8 19)(9 17)(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), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,17,9,13)(2,20,8,15)(3,18,7,12)(4,16,6,14)(5,19,10,11), (1,13)(2,11)(3,14)(4,12)(5,15)(6,18)(7,16)(8,19)(9,17)(10,20)>;`

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

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

`G:=TransitiveGroup(20,57);`

On 25 points: primitive - transitive group 25T21
Generators in S25
```(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)
(1 25 19 14 9)(2 21 20 15 10)(3 22 16 11 6)(4 23 17 12 7)(5 24 18 13 8)
(2 4 5 3)(6 15 23 18)(7 13 22 20)(8 11 21 17)(9 14 25 19)(10 12 24 16)
(2 14)(3 25)(4 9)(5 19)(6 23)(8 17)(10 12)(11 21)(13 20)(16 24)```

`G:=sub<Sym(25)| (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), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (2,4,5,3)(6,15,23,18)(7,13,22,20)(8,11,21,17)(9,14,25,19)(10,12,24,16), (2,14)(3,25)(4,9)(5,19)(6,23)(8,17)(10,12)(11,21)(13,20)(16,24)>;`

`G:=Group( (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), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (2,4,5,3)(6,15,23,18)(7,13,22,20)(8,11,21,17)(9,14,25,19)(10,12,24,16), (2,14)(3,25)(4,9)(5,19)(6,23)(8,17)(10,12)(11,21)(13,20)(16,24) );`

`G=PermutationGroup([[(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)], [(1,25,19,14,9),(2,21,20,15,10),(3,22,16,11,6),(4,23,17,12,7),(5,24,18,13,8)], [(2,4,5,3),(6,15,23,18),(7,13,22,20),(8,11,21,17),(9,14,25,19),(10,12,24,16)], [(2,14),(3,25),(4,9),(5,19),(6,23),(8,17),(10,12),(11,21),(13,20),(16,24)]])`

`G:=TransitiveGroup(25,21);`

D5≀C2 is a maximal subgroup of   D5≀C2⋊C2
D5≀C2 is a maximal quotient of   D52⋊C4  C2.D5≀C2  C52⋊D8  C52⋊SD16  C52⋊Q16

Polynomial with Galois group D5≀C2 over ℚ
actionf(x)Disc(f)
10T19x10+2x9-35x8-80x7+419x6+1106x5-1843x4-6208x3+869x2+11495x+665534·511·116·234·895
10T21x10+2x9-18x8-12x7+66x6+42x5-68x4-52x3+16x2+20x+4218·55·114·1012

Matrix representation of D5≀C2 in GL4(𝔽41) generated by

 0 7 0 0 35 6 0 0 0 0 40 1 0 0 5 35
,
 40 1 0 0 5 35 0 0 0 0 0 7 0 0 35 6
,
 0 0 6 1 0 0 6 35 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 0 0 6 1 0 0 6 35
`G:=sub<GL(4,GF(41))| [0,35,0,0,7,6,0,0,0,0,40,5,0,0,1,35],[40,5,0,0,1,35,0,0,0,0,0,35,0,0,7,6],[0,0,1,0,0,0,0,1,6,6,0,0,1,35,0,0],[1,0,0,0,0,1,0,0,0,0,6,6,0,0,1,35] >;`

D5≀C2 in GAP, Magma, Sage, TeX

`D_5\wr C_2`
`% in TeX`

`G:=Group("D5wrC2");`
`// GroupNames label`

`G:=SmallGroup(200,43);`
`// by ID`

`G=gap.SmallGroup(200,43);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,5,61,2403,408,173,404,109,1014]);`
`// Polycyclic`

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

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