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## G = D10.4D10order 400 = 24·52

### 4th non-split extension by D10 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10.4D10
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — D5×Dic5 — D10.4D10
 Lower central C52 — C5×C10 — D10.4D10
 Upper central C1 — C2 — C22

Generators and relations for D10.4D10
G = < a,b,c,d | a10=b2=1, c10=d2=a5, bab=cac-1=dad-1=a-1, cbc-1=a3b, dbd-1=a8b, dcd-1=c9 >

Subgroups: 508 in 96 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C52, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5⋊D4, C5×D4, C5×D5, C5×C10, C5×C10, D42D5, C5×Dic5, C526C4, D5×C10, C102, D5×Dic5, C522D4, C522Q8, C5×C5⋊D4, C2×C526C4, D10.4D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, D42D5, D52, C2×D52, D10.4D10

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

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

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

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

46 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 ··· 10T 10U 10V 10W 10X 20A 20B 20C 20D 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 size 1 1 2 10 10 10 10 25 25 50 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 20 20 20 20 20 20 20 20

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D4⋊2D5 D52 C2×D52 D10.4D10 kernel D10.4D10 D5×Dic5 C52⋊2D4 C52⋊2Q8 C5×C5⋊D4 C2×C52⋊6C4 C5⋊D4 C52 Dic5 D10 C2×C10 C5 C22 C2 C1 # reps 1 2 1 1 2 1 4 2 4 4 4 4 4 4 8

Matrix representation of D10.4D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 34 0 0 0 0 7 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 7 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 40 34 0 0 0 0 7 7
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 7 1 0 0 0 0 0 0 40 34 0 0 0 0 0 1

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

D10.4D10 in GAP, Magma, Sage, TeX

`D_{10}._4D_{10}`
`% in TeX`

`G:=Group("D10.4D10");`
`// GroupNames label`

`G:=SmallGroup(400,174);`
`// by ID`

`G=gap.SmallGroup(400,174);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=1,c^10=d^2=a^5,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^8*b,d*c*d^-1=c^9>;`
`// generators/relations`

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