Copied to
clipboard

## G = D10⋊D10order 400 = 24·52

### 3rd semidirect product of D10 and D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10⋊D10
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — D10⋊D10
 Lower central C52 — C5×C10 — D10⋊D10
 Upper central C1 — C2 — C22

Generators and relations for D10⋊D10
G = < a,b,c,d | a10=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a5b, dbd=a3b, dcd=c-1 >

Subgroups: 1100 in 140 conjugacy classes, 34 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×2], C22, C22 [×8], C5 [×2], C5 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×16], C10 [×2], C10 [×10], C2×D4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×24], C2×C10 [×2], C2×C10 [×4], C52, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×6], C5×D5 [×2], C5⋊D5 [×2], C5⋊D5, C5×C10, C5×C10, D4×D5 [×2], C5×Dic5 [×2], D52 [×2], D5×C10 [×2], C2×C5⋊D5 [×2], C2×C5⋊D5 [×2], C102, Dic52D5, C5⋊D20 [×2], C5×C5⋊D4 [×2], C2×D52, C22×C5⋊D5, D10⋊D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5 [×2], C2×D4, D10 [×6], C22×D5 [×2], D4×D5 [×2], D52, C2×D52, D10⋊D10

Permutation representations of D10⋊D10
On 20 points - transitive group 20T106
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 12 13 14 15 16 17 18 19 20)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 20)(16 19)(17 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), (1,12)(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)(11,12,13,14,15,16,17,18,19,20), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,20)(16,19)(17,18)>;`

`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)(11,12,13,14,15,16,17,18,19,20), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,20)(16,19)(17,18) );`

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

`G:=TransitiveGroup(20,106);`

46 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E ··· 10T 10U 10V 10W 10X 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 ··· 10 10 10 10 10 20 20 20 20 size 1 1 2 10 10 25 25 50 10 10 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 20 20 20 20 20 20 20 20

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D4×D5 D52 C2×D52 D10⋊D10 kernel D10⋊D10 Dic5⋊2D5 C5⋊D20 C5×C5⋊D4 C2×D52 C22×C5⋊D5 C5⋊D5 C5⋊D4 Dic5 D10 C2×C10 C5 C22 C2 C1 # reps 1 1 2 2 1 1 2 4 4 4 4 4 4 4 8

Matrix representation of D10⋊D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 6 0 0 0 0 35 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 0 40 0 0 0 0 0 0 35 40 0 0 0 0 35 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 34 0 0 0 0 7 40
,
 1 0 0 0 0 0 40 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 7 40 0 0 0 0 7 34

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,35,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,34,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,7,0,0,0,0,40,34] >;`

D10⋊D10 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(400,180);`
`// by ID`

`G=gap.SmallGroup(400,180);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,116,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^5*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽