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## G = C102order 100 = 22·52

### Abelian group of type [10,10]

Aliases: C102, SmallGroup(100,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C102
 Chief series C1 — C5 — C52 — C5×C10 — C102
 Lower central C1 — C102
 Upper central C1 — C102

Generators and relations for C102
G = < a,b | a10=b10=1, ab=ba >

Smallest permutation representation of C102
Regular action 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 32 57 97 11 78 42 64 85 28)(2 33 58 98 12 79 43 65 86 29)(3 34 59 99 13 80 44 66 87 30)(4 35 60 100 14 71 45 67 88 21)(5 36 51 91 15 72 46 68 89 22)(6 37 52 92 16 73 47 69 90 23)(7 38 53 93 17 74 48 70 81 24)(8 39 54 94 18 75 49 61 82 25)(9 40 55 95 19 76 50 62 83 26)(10 31 56 96 20 77 41 63 84 27)```

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

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

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

C102 is a maximal subgroup of   C527D4  C52⋊A4

100 conjugacy classes

 class 1 2A 2B 2C 5A ··· 5X 10A ··· 10BT order 1 2 2 2 5 ··· 5 10 ··· 10 size 1 1 1 1 1 ··· 1 1 ··· 1

100 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C5 C10 kernel C102 C5×C10 C2×C10 C10 # reps 1 3 24 72

Matrix representation of C102 in GL2(𝔽11) generated by

 3 0 0 10
,
 8 0 0 5
`G:=sub<GL(2,GF(11))| [3,0,0,10],[8,0,0,5] >;`

C102 in GAP, Magma, Sage, TeX

`C_{10}^2`
`% in TeX`

`G:=Group("C10^2");`
`// GroupNames label`

`G:=SmallGroup(100,16);`
`// by ID`

`G=gap.SmallGroup(100,16);`
`# by ID`

`G:=PCGroup([4,-2,-2,-5,-5]);`
`// Polycyclic`

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

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