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## G = D10⋊F5order 400 = 24·52

### 2nd semidirect product of D10 and F5 acting via F5/D5=C2

Aliases: D102F5, C5⋊D5.1D4, (D5×C10)⋊2C4, C10.2(C2×F5), C2.3(D5⋊F5), C51(C22⋊F5), C522(C22⋊C4), (C2×D52).2C2, (C5×C10).9(C2×C4), (C2×C52⋊C4)⋊1C2, (C2×C5⋊F5)⋊1C2, (C2×C5⋊D5).4C22, SmallGroup(400,125)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — D10⋊F5
 Lower central C52 — C5×C10 — D10⋊F5
 Upper central C1 — C2

Generators and relations for D10⋊F5
G = < a,b,c,d | a10=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a7b, dcd-1=c3 >

Subgroups: 748 in 84 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2 [×4], C4 [×2], C22 [×5], C5 [×2], C5 [×2], C2×C4 [×2], C23, D5 [×10], C10 [×2], C10 [×4], C22⋊C4, F5 [×8], D10 [×2], D10 [×8], C2×C10 [×2], C52, C2×F5 [×6], C22×D5 [×2], C5×D5 [×2], C5⋊D5 [×2], C5×C10, C22⋊F5 [×2], C5⋊F5, C52⋊C4, D52 [×2], D5×C10 [×2], C2×C5⋊D5, C2×C5⋊F5, C2×C52⋊C4, C2×D52, D10⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5 [×2], C2×F5 [×2], C22⋊F5 [×2], D5⋊F5, D10⋊F5

Character table of D10⋊F5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H size 1 1 10 10 25 25 50 50 50 50 4 4 8 8 4 4 8 8 20 20 20 20 ρ1 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 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 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 linear of order 2 ρ5 1 1 -1 1 -1 -1 -i i -i i 1 1 1 1 1 1 1 1 1 -1 1 -1 linear of order 4 ρ6 1 1 1 -1 -1 -1 i i -i -i 1 1 1 1 1 1 1 1 -1 1 -1 1 linear of order 4 ρ7 1 1 -1 1 -1 -1 i -i i -i 1 1 1 1 1 1 1 1 1 -1 1 -1 linear of order 4 ρ8 1 1 1 -1 -1 -1 -i -i i i 1 1 1 1 1 1 1 1 -1 1 -1 1 linear of order 4 ρ9 2 -2 0 0 2 -2 0 0 0 0 2 2 2 2 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 0 0 -2 2 0 0 0 0 2 2 2 2 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 0 0 0 0 0 0 4 -1 -1 -1 4 -1 -1 -1 0 -1 0 -1 orthogonal lifted from F5 ρ12 4 4 -4 0 0 0 0 0 0 0 4 -1 -1 -1 4 -1 -1 -1 0 1 0 1 orthogonal lifted from C2×F5 ρ13 4 4 0 -4 0 0 0 0 0 0 -1 4 -1 -1 -1 4 -1 -1 1 0 1 0 orthogonal lifted from C2×F5 ρ14 4 4 0 4 0 0 0 0 0 0 -1 4 -1 -1 -1 4 -1 -1 -1 0 -1 0 orthogonal lifted from F5 ρ15 4 -4 0 0 0 0 0 0 0 0 4 -1 -1 -1 -4 1 1 1 0 √5 0 -√5 orthogonal lifted from C22⋊F5 ρ16 4 -4 0 0 0 0 0 0 0 0 -1 4 -1 -1 1 -4 1 1 -√5 0 √5 0 orthogonal lifted from C22⋊F5 ρ17 4 -4 0 0 0 0 0 0 0 0 -1 4 -1 -1 1 -4 1 1 √5 0 -√5 0 orthogonal lifted from C22⋊F5 ρ18 4 -4 0 0 0 0 0 0 0 0 4 -1 -1 -1 -4 1 1 1 0 -√5 0 √5 orthogonal lifted from C22⋊F5 ρ19 8 -8 0 0 0 0 0 0 0 0 -2 -2 3 -2 2 2 2 -3 0 0 0 0 orthogonal faithful ρ20 8 -8 0 0 0 0 0 0 0 0 -2 -2 -2 3 2 2 -3 2 0 0 0 0 orthogonal faithful ρ21 8 8 0 0 0 0 0 0 0 0 -2 -2 3 -2 -2 -2 -2 3 0 0 0 0 orthogonal lifted from D5⋊F5 ρ22 8 8 0 0 0 0 0 0 0 0 -2 -2 -2 3 -2 -2 3 -2 0 0 0 0 orthogonal lifted from D5⋊F5

Permutation representations of D10⋊F5
On 20 points - transitive group 20T101
Generators in S20
```(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)
(2 8 10 4)(3 5 9 7)(11 14 15 12)(13 18)(16 19 20 17)```

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

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

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

`G:=TransitiveGroup(20,101);`

Matrix representation of D10⋊F5 in GL8(ℤ)

 0 0 0 -1 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 0 0 0 0 0 -1 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 0 -1 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 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 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 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())| [0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0],[0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0],[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,-1,0,0,0,0,0,0,0,0,-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] >;`

D10⋊F5 in GAP, Magma, Sage, TeX

`D_{10}\rtimes F_5`
`% in TeX`

`G:=Group("D10:F5");`
`// GroupNames label`

`G:=SmallGroup(400,125);`
`// by ID`

`G=gap.SmallGroup(400,125);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,1444,970,496,8645,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=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^7*b,d*c*d^-1=c^3>;`
`// generators/relations`

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