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## G = D4.D10order 160 = 25·5

### 1st non-split extension by D4 of 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 — C4○D20 — D4.D10
 Lower central C5 — C10 — C20 — D4.D10
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for D4.D10
G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c9 >

Subgroups: 208 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, D5, C10, C10 [×3], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C8⋊C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], C4○D20, D4×C10, D4.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C5⋊D4, D4.D10

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

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

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

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

31 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 2 4 4 4 5 5 8 8 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 2 4 4 20 2 2 20 2 2 20 20 2 ··· 2 4 ··· 4 4 4 4 4

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4.D10 kernel D4.D10 C4.Dic5 D4⋊D5 D4.D5 C4○D20 D4×C10 C20 C2×C10 C2×D4 C2×C4 D4 C4 C22 C5 C1 # reps 1 1 2 2 1 1 1 1 2 2 4 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 40 38 0 0 0 0 28 1 0 0 0 0 0 0 1 3 0 0 0 0 13 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 38 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 28 1
,
 0 6 0 0 0 0 34 7 0 0 0 0 0 0 40 38 0 0 0 0 28 1 0 0 0 0 0 0 40 38 0 0 0 0 28 1
,
 9 16 0 0 0 0 36 32 0 0 0 0 0 0 0 0 40 38 0 0 0 0 28 1 0 0 40 38 0 0 0 0 28 1 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,1,13,0,0,0,0,3,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,0,1],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,40,28,0,0,0,0,38,1,0,0] >;`

D4.D10 in GAP, Magma, Sage, TeX

`D_4.D_{10}`
`% in TeX`

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

`G:=SmallGroup(160,153);`
`// by ID`

`G=gap.SmallGroup(160,153);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,579,159,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^9>;`
`// generators/relations`

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