Copied to
clipboard

## G = D5⋊F5order 200 = 23·52

### The semidirect product of D5 and F5 acting via F5/D5=C2

Aliases: D5⋊F5, D52.2C2, C5⋊F5⋊C2, C51(C2×F5), (C5×D5)⋊2C4, C522(C2×C4), C52⋊C41C2, C5⋊D5.1C22, Hol(D5), SmallGroup(200,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D5⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C5⋊F5 — D5⋊F5
 Lower central C52 — D5⋊F5
 Upper central C1

Generators and relations for D5⋊F5
G = < a,b,c,d | a5=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c3 >

5C2
5C2
25C2
2C5
2C5
25C4
25C22
25C4
5D5
5C10
5C10
5D5
10D5
10D5
25C2×C4
5F5
5F5
5F5
5D10
5F5
5D10
10F5
10F5

Character table of D5⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 10A 10B size 1 5 5 25 25 25 25 25 4 4 8 8 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 1 1 -1 -1 -i -i i i 1 1 1 1 -1 1 linear of order 4 ρ6 1 1 -1 -1 i i -i -i 1 1 1 1 -1 1 linear of order 4 ρ7 1 -1 1 -1 i -i -i i 1 1 1 1 1 -1 linear of order 4 ρ8 1 -1 1 -1 -i i i -i 1 1 1 1 1 -1 linear of order 4 ρ9 4 4 0 0 0 0 0 0 4 -1 -1 -1 0 -1 orthogonal lifted from F5 ρ10 4 -4 0 0 0 0 0 0 4 -1 -1 -1 0 1 orthogonal lifted from C2×F5 ρ11 4 0 -4 0 0 0 0 0 -1 4 -1 -1 1 0 orthogonal lifted from C2×F5 ρ12 4 0 4 0 0 0 0 0 -1 4 -1 -1 -1 0 orthogonal lifted from F5 ρ13 8 0 0 0 0 0 0 0 -2 -2 3 -2 0 0 orthogonal faithful ρ14 8 0 0 0 0 0 0 0 -2 -2 -2 3 0 0 orthogonal faithful

Permutation representations of D5⋊F5
On 10 points - transitive group 10T17
Generators in S10
```(1 2 3 4 5)(6 7 8 9 10)
(1 8)(2 7)(3 6)(4 10)(5 9)
(1 2 3 4 5)(6 10 9 8 7)
(2 3 5 4)(6 8 7 10)```

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

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

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

`G:=TransitiveGroup(10,17);`

On 20 points - transitive group 20T54
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 18)(12 17)(13 16)(14 20)(15 19)
(1 2 3 4 5)(6 10 9 8 7)(11 12 13 14 15)(16 20 19 18 17)
(1 12)(2 14 5 15)(3 11 4 13)(6 17 7 19)(8 16 10 20)(9 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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,18)(12,17)(13,16)(14,20)(15,19), (1,2,3,4,5)(6,10,9,8,7)(11,12,13,14,15)(16,20,19,18,17), (1,12)(2,14,5,15)(3,11,4,13)(6,17,7,19)(8,16,10,20)(9,18)>;`

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

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

`G:=TransitiveGroup(20,54);`

On 25 points - transitive group 25T19
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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 23)(24 25)
(1 19 9 14 25)(2 20 10 15 21)(3 16 6 11 22)(4 17 7 12 23)(5 18 8 13 24)
(2 3 5 4)(6 24 12 20)(7 21 11 18)(8 23 15 16)(9 25 14 19)(10 22 13 17)```

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

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

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

`G:=TransitiveGroup(25,19);`

D5⋊F5 is a maximal subgroup of   F52  C52⋊M4(2)  D5≀C2⋊C2
D5⋊F5 is a maximal quotient of   C523C42  D10⋊F5  Dic5⋊F5  D10.2F5  C524M4(2)

Polynomial with Galois group D5⋊F5 over ℚ
actionf(x)Disc(f)
10T17x10-40x8+515x6-48x5-2600x4+390x3+4225x2-507214·35·510·114·138·9438192

Matrix representation of D5⋊F5 in GL8(ℤ)

 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1
,
 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
,
 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1

`G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0],[-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0],[1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1] >;`

D5⋊F5 in GAP, Magma, Sage, TeX

`D_5\rtimes F_5`
`% in TeX`

`G:=Group("D5:F5");`
`// GroupNames label`

`G:=SmallGroup(200,42);`
`// by ID`

`G=gap.SmallGroup(200,42);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,20,483,328,173,3004,1014]);`
`// Polycyclic`

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

Export

׿
×
𝔽