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## G = C20.30D10order 400 = 24·52

### 4th non-split extension by C20 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20.30D10
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — D5×C20 — C20.30D10
 Lower central C52 — C5×C10 — C20.30D10
 Upper central C1 — C4

Generators and relations for C20.30D10
G = < a,b,c | a20=1, b10=a5, c2=a15, bab-1=cac-1=a9, cbc-1=a10b9 >

Smallest permutation representation of C20.30D10
On 80 points
Generators in S80
```(1 35 29 23 17 11 5 39 33 27 21 15 9 3 37 31 25 19 13 7)(2 28 14 40 26 12 38 24 10 36 22 8 34 20 6 32 18 4 30 16)(41 75 69 63 57 51 45 79 73 67 61 55 49 43 77 71 65 59 53 47)(42 68 54 80 66 52 78 64 50 76 62 48 74 60 46 72 58 44 70 56)
(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)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 71 31 61 21 51 11 41)(2 60 32 50 22 80 12 70)(3 49 33 79 23 69 13 59)(4 78 34 68 24 58 14 48)(5 67 35 57 25 47 15 77)(6 56 36 46 26 76 16 66)(7 45 37 75 27 65 17 55)(8 74 38 64 28 54 18 44)(9 63 39 53 29 43 19 73)(10 52 40 42 30 72 20 62)```

`G:=sub<Sym(80)| (1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7)(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16)(41,75,69,63,57,51,45,79,73,67,61,55,49,43,77,71,65,59,53,47)(42,68,54,80,66,52,78,64,50,76,62,48,74,60,46,72,58,44,70,56), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,71,31,61,21,51,11,41)(2,60,32,50,22,80,12,70)(3,49,33,79,23,69,13,59)(4,78,34,68,24,58,14,48)(5,67,35,57,25,47,15,77)(6,56,36,46,26,76,16,66)(7,45,37,75,27,65,17,55)(8,74,38,64,28,54,18,44)(9,63,39,53,29,43,19,73)(10,52,40,42,30,72,20,62)>;`

`G:=Group( (1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7)(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16)(41,75,69,63,57,51,45,79,73,67,61,55,49,43,77,71,65,59,53,47)(42,68,54,80,66,52,78,64,50,76,62,48,74,60,46,72,58,44,70,56), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,71,31,61,21,51,11,41)(2,60,32,50,22,80,12,70)(3,49,33,79,23,69,13,59)(4,78,34,68,24,58,14,48)(5,67,35,57,25,47,15,77)(6,56,36,46,26,76,16,66)(7,45,37,75,27,65,17,55)(8,74,38,64,28,54,18,44)(9,63,39,53,29,43,19,73)(10,52,40,42,30,72,20,62) );`

`G=PermutationGroup([[(1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7),(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16),(41,75,69,63,57,51,45,79,73,67,61,55,49,43,77,71,65,59,53,47),(42,68,54,80,66,52,78,64,50,76,62,48,74,60,46,72,58,44,70,56)], [(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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,71,31,61,21,51,11,41),(2,60,32,50,22,80,12,70),(3,49,33,79,23,69,13,59),(4,78,34,68,24,58,14,48),(5,67,35,57,25,47,15,77),(6,56,36,46,26,76,16,66),(7,45,37,75,27,65,17,55),(8,74,38,64,28,54,18,44),(9,63,39,53,29,43,19,73),(10,52,40,42,30,72,20,62)]])`

58 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E 5F 5G 5H 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I ··· 20P 20Q 20R 20S 20T 40A ··· 40H order 1 2 2 4 4 4 5 5 5 5 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 10 1 1 10 2 2 2 2 4 4 4 4 10 10 50 50 2 2 2 2 4 4 4 4 10 10 10 10 2 ··· 2 4 ··· 4 10 10 10 10 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 D5 D5 M4(2) Dic5 D10 Dic5 C4×D5 C8⋊D5 C4.Dic5 D52 D5×Dic5 C20.30D10 kernel C20.30D10 C5×C5⋊2C8 C52⋊7C8 D5×C20 C5×Dic5 D5×C10 C5⋊2C8 C4×D5 C52 Dic5 C20 D10 C10 C5 C5 C4 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 2 4 8 8 4 4 8

Matrix representation of C20.30D10 in GL4(𝔽41) generated by

 0 9 0 0 32 13 0 0 0 0 40 0 0 0 0 40
,
 3 0 0 0 18 38 0 0 0 0 32 19 0 0 22 22
,
 28 18 0 0 22 13 0 0 0 0 9 22 0 0 0 32
`G:=sub<GL(4,GF(41))| [0,32,0,0,9,13,0,0,0,0,40,0,0,0,0,40],[3,18,0,0,0,38,0,0,0,0,32,22,0,0,19,22],[28,22,0,0,18,13,0,0,0,0,9,0,0,0,22,32] >;`

C20.30D10 in GAP, Magma, Sage, TeX

`C_{20}._{30}D_{10}`
`% in TeX`

`G:=Group("C20.30D10");`
`// GroupNames label`

`G:=SmallGroup(400,62);`
`// by ID`

`G=gap.SmallGroup(400,62);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,50,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^10=a^5,c^2=a^15,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^9>;`
`// generators/relations`

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