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## G = D10.9D10order 400 = 24·52

### 1st non-split extension by D10 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10.9D10
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C52⋊2D4 — D10.9D10
 Lower central C52 — C5×C10 — D10.9D10
 Upper central C1 — C4

Generators and relations for D10.9D10
G = < a,b,c,d | a10=b2=1, c10=d2=a5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c9 >

Subgroups: 604 in 96 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C2×C20, C5×D5, C5⋊D5, C5×C10, C4○D20, C5×Dic5, C526C4, C5×C20, D5×C10, C2×C5⋊D5, C522D4, C5⋊D20, C522Q8, D5×C20, C4×C5⋊D5, D10.9D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C4○D20, D52, C2×D52, D10.9D10

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

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 20A ··· 20H 20I ··· 20P 20Q ··· 20X order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 10 10 50 1 1 10 10 50 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 ··· 10 2 ··· 2 4 ··· 4 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 C4○D20 D52 C2×D52 D10.9D10 kernel D10.9D10 C52⋊2D4 C5⋊D20 C52⋊2Q8 D5×C20 C4×C5⋊D5 C4×D5 C52 Dic5 C20 D10 C5 C4 C2 C1 # reps 1 1 2 1 2 1 4 2 4 4 4 16 4 4 8

Matrix representation of D10.9D10 in GL4(𝔽41) generated by

 34 7 0 0 34 1 0 0 0 0 1 0 0 0 0 1
,
 14 11 0 0 27 27 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 40 6 0 0 35 35
,
 30 9 0 0 32 11 0 0 0 0 40 6 0 0 0 1
`G:=sub<GL(4,GF(41))| [34,34,0,0,7,1,0,0,0,0,1,0,0,0,0,1],[14,27,0,0,11,27,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,40,35,0,0,6,35],[30,32,0,0,9,11,0,0,0,0,40,0,0,0,6,1] >;`

D10.9D10 in GAP, Magma, Sage, TeX

`D_{10}._9D_{10}`
`% in TeX`

`G:=Group("D10.9D10");`
`// GroupNames label`

`G:=SmallGroup(400,167);`
`// by ID`

`G=gap.SmallGroup(400,167);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,50,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=1,c^10=d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^9>;`
`// generators/relations`

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