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## G = D4⋊6D10order 160 = 25·5

### 2nd semidirect product of D4 and D10 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4⋊6D10
 Chief series C1 — C5 — C10 — D10 — C22×D5 — D4×D5 — D4⋊6D10
 Lower central C5 — C10 — D4⋊6D10
 Upper central C1 — C2 — C2×D4

Generators and relations for D46D10
G = < a,b,c,d | a4=b2=c10=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 520 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4○D20, D4×D5, D42D5, C2×C5⋊D4, D4×C10, D46D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C23×D5, D46D10

Smallest permutation representation of D46D10
On 40 points
Generators in S40
```(1 37 10 32)(2 33 6 38)(3 39 7 34)(4 35 8 40)(5 31 9 36)(11 24 16 29)(12 30 17 25)(13 26 18 21)(14 22 19 27)(15 28 20 23)
(1 23)(2 29)(3 25)(4 21)(5 27)(6 24)(7 30)(8 26)(9 22)(10 28)(11 38)(12 34)(13 40)(14 36)(15 32)(16 33)(17 39)(18 35)(19 31)(20 37)
(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)
(2 5)(3 4)(6 9)(7 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 30)(22 29)(23 28)(24 27)(25 26)(31 33)(34 40)(35 39)(36 38)```

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

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

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

37 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 5A 5B 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 ··· 2 2 2 2 2 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 2 ··· 2 10 10 10 10 2 2 10 10 10 10 2 2 2 ··· 2 4 ··· 4 4 4 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⋊6D10 kernel D4⋊6D10 C4○D20 D4×D5 D4⋊2D5 C2×C5⋊D4 D4×C10 C2×D4 C2×C4 D4 C23 C5 C1 # reps 1 2 4 4 4 1 2 2 8 4 1 4

Matrix representation of D46D10 in GL4(𝔽41) generated by

 40 40 39 25 17 17 0 25 0 40 18 1 17 35 3 7
,
 0 0 34 1 1 1 39 40 0 0 40 0 1 0 34 0
,
 40 7 0 0 34 7 0 0 28 35 35 34 25 33 6 0
,
 40 0 0 0 34 1 0 0 22 23 6 40 31 38 35 35
`G:=sub<GL(4,GF(41))| [40,17,0,17,40,17,40,35,39,0,18,3,25,25,1,7],[0,1,0,1,0,1,0,0,34,39,40,34,1,40,0,0],[40,34,28,25,7,7,35,33,0,0,35,6,0,0,34,0],[40,34,22,31,0,1,23,38,0,0,6,35,0,0,40,35] >;`

D46D10 in GAP, Magma, Sage, TeX

`D_4\rtimes_6D_{10}`
`% in TeX`

`G:=Group("D4:6D10");`
`// GroupNames label`

`G:=SmallGroup(160,219);`
`// by ID`

`G=gap.SmallGroup(160,219);`
`# by ID`

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

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

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