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## G = C20.29D10order 400 = 24·52

### 3rd non-split extension by C20 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C20.29D10
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C5×C5⋊2C8 — C20.29D10
 Lower central C52 — C20.29D10
 Upper central C1 — C4

Generators and relations for C20.29D10
G = < a,b,c | a20=1, b10=a5, c2=a10, bab-1=cac-1=a9, cbc-1=b9 >

Subgroups: 364 in 60 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2×C4, D5, C10, C10, C2×C8, Dic5, C20, C20, D10, C52, C52C8, C40, C4×D5, C5⋊D5, C5×C10, C8×D5, C526C4, C5×C20, C2×C5⋊D5, C5×C52C8, C4×C5⋊D5, C20.29D10
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C2×C8, D10, C4×D5, C8×D5, D52, Dic52D5, C20.29D10

Smallest permutation representation of C20.29D10
On 40 points
Generators in S40
```(1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39)(2 20 38 16 34 12 30 8 26 4 22 40 18 36 14 32 10 28 6 24)
(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 15 21 35)(2 24 22 4)(3 33 23 13)(5 11 25 31)(6 20 26 40)(7 29 27 9)(8 38 28 18)(10 16 30 36)(12 34 32 14)(17 39 37 19)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E 5F 5G 5H 8A ··· 8H 10A 10B 10C 10D 10E 10F 10G 10H 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 4 4 4 4 5 5 5 5 5 5 5 5 8 ··· 8 10 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 25 25 1 1 25 25 2 2 2 2 4 4 4 4 5 ··· 5 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 D5 D10 C4×D5 C8×D5 D52 Dic5⋊2D5 C20.29D10 kernel C20.29D10 C5×C5⋊2C8 C4×C5⋊D5 C52⋊6C4 C2×C5⋊D5 C5⋊D5 C5⋊2C8 C20 C10 C5 C4 C2 C1 # reps 1 2 1 2 2 8 4 4 8 16 4 4 8

Matrix representation of C20.29D10 in GL4(𝔽41) generated by

 0 32 0 0 9 22 0 0 0 0 1 0 0 0 0 1
,
 14 0 0 0 25 27 0 0 0 0 40 6 0 0 35 35
,
 9 0 0 0 19 32 0 0 0 0 40 6 0 0 0 1
`G:=sub<GL(4,GF(41))| [0,9,0,0,32,22,0,0,0,0,1,0,0,0,0,1],[14,25,0,0,0,27,0,0,0,0,40,35,0,0,6,35],[9,19,0,0,0,32,0,0,0,0,40,0,0,0,6,1] >;`

C20.29D10 in GAP, Magma, Sage, TeX

`C_{20}._{29}D_{10}`
`% in TeX`

`G:=Group("C20.29D10");`
`// GroupNames label`

`G:=SmallGroup(400,61);`
`// by ID`

`G=gap.SmallGroup(400,61);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,31,50,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^10=a^5,c^2=a^10,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=b^9>;`
`// generators/relations`

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