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## G = D4⋊8D10order 160 = 25·5

### 4th semidirect product of D4 and D10 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4⋊8D10
 Chief series C1 — C5 — C10 — D10 — C22×D5 — D4×D5 — D4⋊8D10
 Lower central C5 — C10 — D4⋊8D10
 Upper central C1 — C2 — C4○D4

Generators and relations for D48D10
G = < a,b,c,d | a4=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 568 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, 2+ 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×D20, C4○D20, D4×D5, Q82D5, C5×C4○D4, D48D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C23×D5, D48D10

Smallest permutation representation of D48D10
On 40 points
Generators in S40
```(1 21 33 19)(2 22 34 20)(3 23 35 11)(4 24 36 12)(5 25 37 13)(6 26 38 14)(7 27 39 15)(8 28 40 16)(9 29 31 17)(10 30 32 18)
(1 19)(2 22)(3 11)(4 24)(5 13)(6 26)(7 15)(8 28)(9 17)(10 30)(12 36)(14 38)(16 40)(18 32)(20 34)(21 33)(23 35)(25 37)(27 39)(29 31)
(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 13)(2 12)(3 11)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)```

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

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

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

37 conjugacy classes

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

37 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 D10 D10 D10 2+ 1+4 D4⋊8D10 kernel D4⋊8D10 C2×D20 C4○D20 D4×D5 Q8⋊2D5 C5×C4○D4 C4○D4 C2×C4 D4 Q8 C5 C1 # reps 1 3 3 6 2 1 2 6 6 2 1 4

Matrix representation of D48D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 13 0 0 5 0 0 7 0 40 28 0 0 0 1 0 40 0 0 7 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 5 0 0 7 0 40 28 0 0 3 40 0 40 0 0 32 0 0 28
,
 7 7 0 0 0 0 34 40 0 0 0 0 0 0 1 0 5 0 0 0 0 0 28 1 0 0 0 0 40 0 0 0 0 1 28 0
,
 7 7 0 0 0 0 40 34 0 0 0 0 0 0 13 0 0 5 0 0 32 0 1 28 0 0 0 1 0 40 0 0 32 0 0 28

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,13,7,0,7,0,0,0,0,1,0,0,0,0,40,0,0,0,0,5,28,40,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,7,3,32,0,0,0,0,40,0,0,0,0,40,0,0,0,0,5,28,40,28],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,28,40,28,0,0,0,1,0,0],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,13,32,0,32,0,0,0,0,1,0,0,0,0,1,0,0,0,0,5,28,40,28] >;`

D48D10 in GAP, Magma, Sage, TeX

`D_4\rtimes_8D_{10}`
`% in TeX`

`G:=Group("D4:8D10");`
`// GroupNames label`

`G:=SmallGroup(160,224);`
`// by ID`

`G=gap.SmallGroup(160,224);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,579,69,4613]);`
`// 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,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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