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## G = C50.C10order 500 = 22·53

### The non-split extension by C50 of C10 acting faithfully

Aliases: C50.C10, C252C20, Dic25⋊C5, C52.Dic5, 5- 1+22C4, C2.(C25⋊C10), C10.3(C5×D5), (C5×C10).2D5, C5.3(C5×Dic5), (C2×5- 1+2).C2, SmallGroup(500,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C50.C10
 Chief series C1 — C5 — C25 — C50 — C2×5- 1+2 — C50.C10
 Lower central C25 — C50.C10
 Upper central C1 — C2

Generators and relations for C50.C10
G = < a,b | a50=1, b10=a25, bab-1=a9 >

Smallest permutation representation of C50.C10
On 100 points
Generators in S100
```(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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 62 26 87)(2 51 47 56 42 61 37 66 32 71 27 76 22 81 17 86 12 91 7 96)(3 90 18 75 33 60 48 95 13 80 28 65 43 100 8 85 23 70 38 55)(4 79 39 94 24 59 9 74 44 89 29 54 14 69 49 84 34 99 19 64)(5 68 10 63 15 58 20 53 25 98 30 93 35 88 40 83 45 78 50 73)(6 57 31 82)(11 52 36 77)(16 97 41 72)(21 92 46 67)```

`G:=sub<Sym(100)| (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,62,26,87)(2,51,47,56,42,61,37,66,32,71,27,76,22,81,17,86,12,91,7,96)(3,90,18,75,33,60,48,95,13,80,28,65,43,100,8,85,23,70,38,55)(4,79,39,94,24,59,9,74,44,89,29,54,14,69,49,84,34,99,19,64)(5,68,10,63,15,58,20,53,25,98,30,93,35,88,40,83,45,78,50,73)(6,57,31,82)(11,52,36,77)(16,97,41,72)(21,92,46,67)>;`

`G:=Group( (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,62,26,87)(2,51,47,56,42,61,37,66,32,71,27,76,22,81,17,86,12,91,7,96)(3,90,18,75,33,60,48,95,13,80,28,65,43,100,8,85,23,70,38,55)(4,79,39,94,24,59,9,74,44,89,29,54,14,69,49,84,34,99,19,64)(5,68,10,63,15,58,20,53,25,98,30,93,35,88,40,83,45,78,50,73)(6,57,31,82)(11,52,36,77)(16,97,41,72)(21,92,46,67) );`

`G=PermutationGroup([[(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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,62,26,87),(2,51,47,56,42,61,37,66,32,71,27,76,22,81,17,86,12,91,7,96),(3,90,18,75,33,60,48,95,13,80,28,65,43,100,8,85,23,70,38,55),(4,79,39,94,24,59,9,74,44,89,29,54,14,69,49,84,34,99,19,64),(5,68,10,63,15,58,20,53,25,98,30,93,35,88,40,83,45,78,50,73),(6,57,31,82),(11,52,36,77),(16,97,41,72),(21,92,46,67)]])`

44 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 10A 10B 10C 10D 10E 10F 20A ··· 20H 25A ··· 25J 50A ··· 50J order 1 2 4 4 5 5 5 5 5 5 10 10 10 10 10 10 20 ··· 20 25 ··· 25 50 ··· 50 size 1 1 25 25 2 2 5 5 5 5 2 2 5 5 5 5 25 ··· 25 10 ··· 10 10 ··· 10

44 irreducible representations

 dim 1 1 1 1 1 1 10 10 2 2 2 2 type + + + - + - image C1 C2 C4 C5 C10 C20 C25⋊C10 C50.C10 D5 Dic5 C5×D5 C5×Dic5 kernel C50.C10 C2×5- 1+2 5- 1+2 Dic25 C50 C25 C2 C1 C5×C10 C52 C10 C5 # reps 1 1 2 4 4 8 2 2 2 2 8 8

Matrix representation of C50.C10 in GL10(𝔽101)

 79 79 21 21 80 80 23 44 0 0 1 1 22 22 79 79 78 21 0 0 80 80 21 21 79 79 0 0 23 44 79 79 22 22 1 1 0 0 78 21 79 100 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 22 0 1 0 0 0 22 0 80 0 21 79 78 0 0 0 21 0 58 0 100 0 0 0 1 0 21 0 80 0 22 0 0 79 78 0 43 0 58
,
 26 80 0 0 0 0 0 0 0 0 13 75 0 0 0 0 0 0 0 0 30 30 71 71 17 17 0 0 9 63 43 43 58 58 21 21 0 0 93 54 88 88 17 17 84 84 30 4 0 0 9 9 97 97 4 4 12 75 0 0 80 54 34 34 62 88 26 97 9 43 58 24 60 60 76 63 34 88 4 64 84 50 26 43 75 75 8 92 9 43 71 11 51 55 67 67 26 97 4 64

`G:=sub<GL(10,GF(101))| [79,1,80,79,79,1,0,0,0,0,79,1,80,79,100,0,22,21,100,22,21,22,21,22,0,0,0,79,0,0,21,22,21,22,0,0,1,78,0,0,80,79,79,1,0,0,0,0,0,79,80,79,79,1,0,0,0,0,1,78,23,78,0,0,0,0,0,0,0,0,44,21,0,0,0,0,22,21,21,43,0,0,23,78,0,0,0,0,0,0,0,0,44,21,0,0,80,58,80,58],[26,13,30,43,88,9,80,58,84,71,80,75,30,43,88,9,54,24,50,11,0,0,71,58,17,97,34,60,26,51,0,0,71,58,17,97,34,60,43,55,0,0,17,21,84,4,62,76,75,67,0,0,17,21,84,4,88,63,75,67,0,0,0,0,30,12,26,34,8,26,0,0,0,0,4,75,97,88,92,97,0,0,9,93,0,0,9,4,9,4,0,0,63,54,0,0,43,64,43,64] >;`

C50.C10 in GAP, Magma, Sage, TeX

`C_{50}.C_{10}`
`% in TeX`

`G:=Group("C50.C10");`
`// GroupNames label`

`G:=SmallGroup(500,9);`
`// by ID`

`G=gap.SmallGroup(500,9);`
`# by ID`

`G:=PCGroup([5,-2,-5,-2,-5,-5,50,3603,1208,418,10004]);`
`// Polycyclic`

`G:=Group<a,b|a^50=1,b^10=a^25,b*a*b^-1=a^9>;`
`// generators/relations`

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