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## G = C110order 110 = 2·5·11

### Cyclic group

Aliases: C110, also denoted Z110, SmallGroup(110,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C110
 Chief series C1 — C11 — C55 — C110
 Lower central C1 — C110
 Upper central C1 — C110

Generators and relations for C110
G = < a | a110=1 >

Smallest permutation representation of C110
Regular action on 110 points
Generators in S110
`(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 99 100 101 102 103 104 105 106 107 108 109 110)`

`G:=sub<Sym(110)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)])`

C110 is a maximal subgroup of   Dic55

110 conjugacy classes

 class 1 2 5A 5B 5C 5D 10A 10B 10C 10D 11A ··· 11J 22A ··· 22J 55A ··· 55AN 110A ··· 110AN order 1 2 5 5 5 5 10 10 10 10 11 ··· 11 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C5 C10 C11 C22 C55 C110 kernel C110 C55 C22 C11 C10 C5 C2 C1 # reps 1 1 4 4 10 10 40 40

Matrix representation of C110 in GL1(𝔽331) generated by

 282
`G:=sub<GL(1,GF(331))| [282] >;`

C110 in GAP, Magma, Sage, TeX

`C_{110}`
`% in TeX`

`G:=Group("C110");`
`// GroupNames label`

`G:=SmallGroup(110,6);`
`// by ID`

`G=gap.SmallGroup(110,6);`
`# by ID`

`G:=PCGroup([3,-2,-5,-11]);`
`// Polycyclic`

`G:=Group<a|a^110=1>;`
`// generators/relations`

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