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G = D4⋊D10order 160 = 25·5

2nd semidirect product of D4 and D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — D4⋊D10
 Lower central C5 — C10 — C20 — D4⋊D10
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4⋊D10
G = < a,b,c,d | a4=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 256 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C8⋊C22, C52C8, D20, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, D4⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8⋊C22, C5⋊D4, C22×D5, C2×C5⋊D4, D4⋊D10

Smallest permutation representation of D4⋊D10
On 40 points
Generators in S40
```(1 11 6 19)(2 12 7 20)(3 13 8 16)(4 14 9 17)(5 15 10 18)(21 32 26 37)(22 33 27 38)(23 34 28 39)(24 35 29 40)(25 36 30 31)
(1 38)(2 34)(3 40)(4 36)(5 32)(6 33)(7 39)(8 35)(9 31)(10 37)(11 27)(12 23)(13 29)(14 25)(15 21)(16 24)(17 30)(18 26)(19 22)(20 28)
(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 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)
(1 17)(2 16)(3 20)(4 19)(5 18)(6 14)(7 13)(8 12)(9 11)(10 15)(22 30)(23 29)(24 28)(25 27)(31 38)(32 37)(33 36)(34 35)(39 40)```

`G:=sub<Sym(40)| (1,11,6,19)(2,12,7,20)(3,13,8,16)(4,14,9,17)(5,15,10,18)(21,32,26,37)(22,33,27,38)(23,34,28,39)(24,35,29,40)(25,36,30,31), (1,38)(2,34)(3,40)(4,36)(5,32)(6,33)(7,39)(8,35)(9,31)(10,37)(11,27)(12,23)(13,29)(14,25)(15,21)(16,24)(17,30)(18,26)(19,22)(20,28), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,17)(2,16)(3,20)(4,19)(5,18)(6,14)(7,13)(8,12)(9,11)(10,15)(22,30)(23,29)(24,28)(25,27)(31,38)(32,37)(33,36)(34,35)(39,40)>;`

`G:=Group( (1,11,6,19)(2,12,7,20)(3,13,8,16)(4,14,9,17)(5,15,10,18)(21,32,26,37)(22,33,27,38)(23,34,28,39)(24,35,29,40)(25,36,30,31), (1,38)(2,34)(3,40)(4,36)(5,32)(6,33)(7,39)(8,35)(9,31)(10,37)(11,27)(12,23)(13,29)(14,25)(15,21)(16,24)(17,30)(18,26)(19,22)(20,28), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,17)(2,16)(3,20)(4,19)(5,18)(6,14)(7,13)(8,12)(9,11)(10,15)(22,30)(23,29)(24,28)(25,27)(31,38)(32,37)(33,36)(34,35)(39,40) );`

`G=PermutationGroup([[(1,11,6,19),(2,12,7,20),(3,13,8,16),(4,14,9,17),(5,15,10,18),(21,32,26,37),(22,33,27,38),(23,34,28,39),(24,35,29,40),(25,36,30,31)], [(1,38),(2,34),(3,40),(4,36),(5,32),(6,33),(7,39),(8,35),(9,31),(10,37),(11,27),(12,23),(13,29),(14,25),(15,21),(16,24),(17,30),(18,26),(19,22),(20,28)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40)], [(1,17),(2,16),(3,20),(4,19),(5,18),(6,14),(7,13),(8,12),(9,11),(10,15),(22,30),(23,29),(24,28),(25,27),(31,38),(32,37),(33,36),(34,35),(39,40)]])`

31 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J order 1 2 2 2 2 2 4 4 4 5 5 8 8 10 10 10 ··· 10 20 20 20 20 20 ··· 20 size 1 1 2 4 20 20 2 2 4 2 2 20 20 2 2 4 ··· 4 2 2 2 2 4 ··· 4

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4⋊D10 kernel D4⋊D10 C4.Dic5 D4⋊D5 Q8⋊D5 C2×D20 C5×C4○D4 C20 C2×C10 C4○D4 C2×C4 D4 Q8 C4 C22 C5 C1 # reps 1 1 2 2 1 1 1 1 2 2 2 2 4 4 1 4

Matrix representation of D4⋊D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1 0 0 0 0 40 0 0 0
,
 1 34 0 0 0 0 7 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 34 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

D4⋊D10 in GAP, Magma, Sage, TeX

`D_4\rtimes D_{10}`
`% in TeX`

`G:=Group("D4:D10");`
`// GroupNames label`

`G:=SmallGroup(160,170);`
`// by ID`

`G=gap.SmallGroup(160,170);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,579,159,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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